PUPILS’  OUTLINES  FOR 


HOME  STUDY 


IN  CONNECTION  WITH  SCHOOL  WORK 


ARITHMETIC,  PART  I 

Integers,  Decimals,  Fractions,  Denominate  Numbers 

By  D.  E.  AXELSTROM 


Price , Fifteen  Cents 


Jennings  Publishing  Company 
P.  O.  Box  17,  Brooklyn,  N.  Y. 


Copyrighted  1911  by  Jennings  Publishing  Co. 


2 


ARITHMETIC-PART  I 


ARITHMETIC 

This  is  the  science  of  numbers  showing  their  various  properties  and  the  art  of 
computing  with  figures,  that  is  applying  the  properties  of  numbers  to  practical 
examples  arising  in  business  or  otherwise. 

Arithmetic  cultivates  observation,  imagination,  reasoning,  accuracy,  and  clear- 
ness of  expression. 

Number  is  a unit  or  collection  of  units  or  a part  of  a unit. 

Integers  are  whole  numbers. 


NOTATION  AND  NUMERATION 

I.  Notation  is  representing  numbers  by  characters  which  may  be  letters,  figures,  or 
words ; as, 

Roman  Notation — by  Letters — V 
Arabic  Notation — by  Figures — 5 
Language — by  Words — five 

(a)  Roman  System  of  Notation — commonly  used  before  the  Arabic  system,  con- 

sists of  7 letters — 

1 = 1 V = 5 X = 10  L = 50  C = 100  D = 500  M = 1000 

Rules: 

1.  If  the  same  letter  or  one  of  less  value  follows  another,  add  the  values  of  both, 

as  XX=10+ 10  or  20.  XV=10+5  or  15. 

2.  If  a letter  of  greater  value  follows  another  subtract  the  value  of  the  less  from 
that  of  the  greater;  as,  XC=10  subtracted  from  100  or  90. 

3.  A bar  over  a letter  multiplies  it  by  a thousand;  as,  V = 5 x 1000  or  5000. 
The  same  letter  should  not  be  repeated  more  than  3 times  in  succession;  as,  4 use 

IV  not  IIII ; for  900  use  CM  not  DCCCC . 

(b)  Arabic  System  of  Notation. — By  the  16th  century,  this  had  come  to  be  gen- 
erally used  among  most  civilized  nations.  It  originated  with  the  Arabs  of 
India  some  2000  years  ago.  It  consisted  of  the  nine  characters  called 
digits — 1,  2,  3,  4,  5,  6,  7,  8,  9,  but  later  the  0 was  added  making  ten 
characters. 

It  is  also  known  as  the  decimal  system  meaning  that  the  increase  and  decrease  is 
tenfold. 

The  place  value  of  the  figures,  makes  it  valuable,  for  instance  in  5 51 500 

the  same  figure  5 has  different  values  according  to  the  place  it  occupies:  in  the  first, 
it  is  5 units;  in  the  second  number  the  5 means  5 tens  or  50+1=51,  etc. ; whereas  in 
the  Roman  system  to  show  5 and  51  we  must  use  entirely  different  characters  to 
indicate  each  5,  for  the  numerals  have  no  place  value;  as,  V=5  and  L 1=51. 

The  Arabic  System  is  a great  improvement  on  the  Roman  system  as  is  shown  by 
multiplying  in  each  system. 


3 ns 

u.  ^ 


ARITHMETIC-PART  I 


3 


II.  Numeration  is  reading  numbers. 

TABLE  OF  NUMERATION 

Beginning  at  the  right  a figure  in  the 


1st  place 

is 

called 

units 

1 

2nd 

i 4 

i i 

tens 

units  period. 

3rd 

i 4 

i < 

4 ( 

hundreds 

J 

4th 

4 4 

< t 

4 4 

thousands 

5th 

i 4 

< * 

4 4 

ten-thousands 

y thousands  period. 

6th 

< ( 

* < 

4 4 

hundred-thousands 

J 

7th 

( < 

4 4 

4 4 

millions 

- i 

8th 

4 4 

4 4 

4 4 

ten-millions 

[ millions  period. 

9th 

4 < 

4 4 

4 4 

hundred-millions 

J 

10th 

1 4 

.4  4 

“ 

billions 

i 

Uth 

i 4 

4 4 

4 4 

ten-billions 

> billions  period. 

12th 

< * 

4 4 

4 4 

hundred-billions 

1 

J 

13th 

4 4 

4 4 

trillions 

14  th 

4 4 

4 4 

4 4 

ten-trillions 

y trillions  period. 

15th 

4 4 

4 4 

4 4 

hundred-trillions 

1 

J 

For  ease  in  reading  and  writing  numbers,  the  different  places  have  been  grouped 
in  threes,  called  periods,  as  shown  above,  and  separated  from  each  other  by  commas ; 
as,  67,894,325. 

Rule.— Numeration: 

A.  Separate  into  the  periods.  Begin  at  right,  count  off  3 places,  and  place  a 
comma  after  each  set  of  3’s  except  the  last. 

B . Begin  at  the  left  and  read  the  figures  in  each  period  as  though  they  stood  alone ; 
and  then  add  the  name  of  that  period ; as,  67  million,  894  thousand,  325  units ; 
but  the  name  units  is  omitted  after  the  last  period. 

Rule,— Notation: 

A.  Begin  at  the  left  and  write  each  period  in  order,  placing  commas  after  each, 
filling  in  the  3 figures  required  for  each  period ; if  any  are  omitted  supply 
ciphers ; as,  in  37  million  506,  write  37  with  a comma  after  it  for  millions — the 
next  period  in  order  towards  the  right  is  thousands;  but  no  figures  are  given 
for  thousands,  so  supply  3 ciphers  for  that  period  and  place  a comma  after 
it— the  next  period  in  order  is  units  and  we  have  506  giving  us  37,000,506. 

DECIMALS 

The  period  is  used  to  indicate  a decimal  and  to  separate  this  part  from  the  whole 
number. 

The  places  right  of  the  decimal  point  are  the  decimal  places,  the  first  being 
tenths,  the  others  following  in  the  same  order  as  for  the  whole  numbers,  but  each 
word  ends  in  ths. 

Numeration — Read  the  decimal  as  though  it  were  a whole  number  and  then  give 
it  the  name  of  the  farthest  right  hand  place;  as,  .0409  would  be  409  ten-thousandths — 
0 being  in  tenths  place,  4 in  hundredths,  0 in  thousandths,  and  9 in  ten-thousandths 
place. 


4 


ARITHMETIC-PART  I 


1.  If  we  start  with  unit  1,  it  will  take  10  of  those  to  make  1 for  the  next  place 
or  1 ten  ; as,  10. 

10.  The  point  or  period  is  a mark  of  separation  between  the  whole  and  its 
parts. 

1.  Starting  with  the  unit  one  again,  if  we  divide  it  by  10  we  will  get  ^ or  1 
tenth  giving  us 

. 1 the  first  decimal  place  or  part.  Proof — it  takes  10  of  the  . 1 to  make  one  of 
the  next  higher  order  or  1 unit. 

Rules: 

A.  Ten  of  any  order,  whether  whole  or  decimal,  makes  1 of  the  next  higher 
order. 

B.  Moving  the  decimal  point  1 place  to  the  right  multiplies  the  number  by  10;  2 
places,  by  100;  3,  by  1000,  and  etc. 

C.  Moving  the  decimal  point  1 place  to  the  left  divides  the  number  by  10,  etc. 

D.  If  we  annex  ciphers  to  the  decimal  places  we  do  not  change  the  value  as  each 
figure  remains  in  the  same  place. 

E.  If  we  prefix  a cipher  in  the  decimal  before  the  first  figure,  we  divide  it  by  10 
for  that  moves  each  figure  one  place  to  the  right  and  is  therefore  only  Jg  as 
great  as  before;  as  .1 — .01 — one  now  is  in  hundredths  place  and  is  only  ^ as 
great  as  it  was  before. 

ADDITION 

It  is  the  process  of  finding  one  number  which  is  equivalent  to  two  or  more  num- 
bers taken  together,  and  the  answer  obtained  is  called  the  sum  or  amount. 

Addends — the  numbers  combined  to  give  the  sum. 

Sign  of  addition  is  read  plus. 

Rules: 

A.  Only  numbers  of  the  same  kind  can  be  added. 

B.  It  makes  no  difference  in  what  order  the  numbers  are  combined. 

C.  If  a column  comes  to  9 or  less  put  it  down  and  there  is  nothing  to  add  to  the 
next  column  but  if  it  comes  to  10  or  more  put  down  the  units  figure  only  and 
add  the  tens  to  the  next  column, 

Proof — Add  each  column  in  the  reverse  order. 

Method — Write  the  numbers  so  that  all  figures  of  the  same  order  will  be  in  the  same 
column,  for  instance  all  units  must  come  in  the  same  column,  all  tenths  in 
the  same  column,  etc. 

When  numbers  have  all  been  written  in  columns  one  under  the  other  begin 
at  the  right  to  add.  In  the  example  given,  5 and  5 are  10,  put  down  the  0 
9 . 25  and  add  the  1 to  the  next  column,  for  the  10  of  any  order  makes  1 of  the  next,  so 

2.75  we  have  none  left  for  units  order  but  one  for  the  next  order.  Add  the  next 

.80  column  and  we  get  18,  which  is  the  same  as  10-f-8,  the  10  will  equal  1 of  the 

next  order  so  we  add  1 to  the  next  column  and  put  down  the  8 which  we  had 

12.80  left  for  the  second  column.  As  soon  as  the  decimal  point  is  reached  put  it 

down  in  the  answer.  Add  the  third  column  and  we  have  12;  following  the 
directions  above  till  finished  we  have  12.80  for  the  answer  or  sum. 

If  a $ occurs  at  the  top  of  the  column  all  the  numbers  in  that  column  are  considered 
dollars. 

In  adding  it  is  easier  to  group  the  digits  in  convenient  combinations  rather  than  to 
add  one  number  after  the  other  in  the  order  they  occur. 


Arithmetic— part  i 


5 


SUBTRACTION 

The  process  of  finding  how  much  one  number  exceeds  another  or  the  difference 
between  two  numbers  is  subtraction. 

The  sign  of  subtraction  is  — , read  minus,  and  shows  that  the  number  following 
the  sign  is  to  be  taken  away  from  the  number  preceding  it. 

Minuend — is  the  larger  number  from  which  we  subtract. 

Subtrahend  is  the  smaller  number  which  is  to  be  subtracted. 

The  difference  or  remainder  is  the  answer. 

Rule — Numbers  subtracted  must  be  of  the  same  kind,  and  the  answer  will  be  the 
same  kind. 

Proof — The  difference  added  to  the  smaller  number,  or  subtrahend,  should  equal 
the  larger  number,  or  minueud. 

Principle  — If  we  add  or  subtract  the  same  number  from  both  minuend  and  sub- 
trahend the  answer  or  difference  is  not  changed. 

Method  — Write  numbers  under  each  other  keeping  the  figures  of  one  order  in  the 
same  column,  and  decimal  points,  if  any,  in  a straight  vertical  line,  the  same  as  in 
addition. 

52.4  Begin  at  the  right.  In  the  example  given,  3 tenths  from  4 tenths  gives  1 

17.3  tenth— put  down  the  decimal  point.  We  cannot  take  7 units  from  2 units 

so  we  take  1 ten  from  the  five  leaving  that  4 and  add  the  1 ten  making  10 

35.1  units  to  the  2,  giving  12  units,  then  7 from  12  is  5' units  and  1 ten  from  4 

tens  is  3 tens  or  35.1,  the  Difference. 

AUSTRIAN  METHOD 

This  is  sometimes  known  as  the  Additive  Method  of  Subtraction  for  we  find  what 
number  added  to  the  subtrahend  will  produce  the  minuend.  This  depends  on  the 
principle  given  above. 

52.4  Beginning  at  the  right  we  find  that  .1  added  to  .3  will  give  the  minuend  .4, 

17.3  so  we  write  .1  in  tenth’s  column,  5 added  to  7 gives  12  so  write  the  5 in 

units  column  and  carry  the  one  ten  to  the  next  column — for  we  added  10 

35.1  to  the  minuend  to  make  12,  therefore  according  to  the  above  principle  we 

must  add  one  ten  to  the  subtrahend  also.  3 added  to  one,  plus  the  one 
ten  we  had  to  carry  gives  5,  so  we  write  3 in  tens  place. 

MULTIPLICATION 

Is  the  process  of  taking  a number  as  many  times  as  there  are  units  in  another. 
The  result  obtained  is  the  product. 

Multiplicand  is  the  number  to  be  taken  or  multiplied. 

Multiplier  is  the  number  by  which  we  multiply  and  shows  how  many  times  the 
other  number  is  to  be  taken. 

The  sign  is  X.  and  is  read  times  when  it  follows  the  multiplier,  or  multiplied  by 
when  it  precedes  the  multiplier. 

Factors  are  the  numbers  which  multiplied  together  produce  a given  number. 
Concrete  Number  is  one  with  a name;  that  is,  the  number  refers  to  some  particular 
object;  as,  7 books. 

Abstract  Number  is  one  without  a name ; that  is,  the  number  does  not  refer  to 
any  particular  object;  as  7. 


6 


ARITHMETIC-PART  I 


Principles: 

1.  Multiplicand  and  the  Product  will  be  of  the  same  kind  or  have  the  same  name, 
if  there  is  any  name. 

2.  Multiplier  should  always  be  considered  abstract,  without  a name  or  without 
being  of  a certain  kind. 

3.  The  figures  in  the  product  will  always  be  the  same  no  matter  which  number  is 
taken  as  the  multiplier. 

Method— Write  the  multiplier  under  the  multiplicand.  Begin  at  the  right  to 
537  multiply.  7 X 7=49.  Put  down  the  9 units  and  add  the  4 tens  to  the 

X 27  next  product.  7X3  tens=21  tens  plus  the  4 tens  gives  25  tens  or  5 tens 

and  2 hundreds.  Put  down  the  5 in  tens’  place  and  add  the  2 hundreds 

3759  to  the  next  product.  7X5  hundreds  plus  2 hundreds=37  hundreds  or  7 

1074  hundreds  and  3 thousands  and,  being  the  last  multiplication,  write  the  7 

in  hundreds’  place  and  the  3 in  thousands’  place.  This  gives  3769  as 

14,499  our  partial  product. 

But  we  had  27  as  a multiplier  giving  7 units  and  2 tens.  Having  gotten  the  par- 
tial product  by  7,  we  multiply  same  as  before  by  2 and  obtain  1074,  but  the  2 stands 
for  tens,  so  our  partial  product  must  be  1074  tens. 

Adding  our  partial  products,  we  have  the  entire  product,  14,499. 

Rule — The  first  right-hand  figure  in  any  partial  product  should  be  placed  in  the 
same  column  directly  under  the  figure  being  used  as  the  multiplier,  z.  e. , multiplying 
by  a figure  in  thousands’  place,  the  first  figure  in  that  partial  product  should  be  placed 
in  the  third  column  or  under  the  thousands  figure. 

Ciphers  at  the  End  of  the  Factors 

If  either  multiplier  or  multiplicand,  or  both,  have  ciphers  at  the  right  of  the 
number,  multiply  the  numbers  as  though  there  were  no  ciphers  and  in  the  answer 
annex  at  the  right  as  many  ciphers  as  there  were  ciphers  in  both  factors. 

Adding  1 cipher  multiplies  by  10 ; adding  2 ciphers  multiplies  by  100,  etc. 

1500  X 210  = ? 


15100 

2ll0 

15 

30 


315,000  Ans. 


Multiplication  of  Decimals 

Multiply  as  though  whole  numbers  were  given,  then  in  the  answer  begin  at  the 
right  and  count  off  as  many  decimal  places  as  there  are  decimal  places  in  both  fac- 
tors and  there  place  a decimal  point.  If  there  are  not  sufficient  figures,  supply 
ciphers. 

1.5  15  by  21  gives  315  and  one  decimal  place  in  the  multiplicand  plus  three 

•021  in  the  multiplier  gives  4 decimal  places  for  the  answer.  Not  having  suf- 

15  ficient  figures  beginning  at  the  right  to  count,  supply  one  cipher  and  put 

— — . down  the  decimal  point. 

.0315 

15  .2  is  the  same  as  2 X -1 ; hence  multiplying  by  2 we  get  30,  and  then  by  .1 

. 2 we  move  the  decimal  point  1 place  to  the  left. 

iTo 


Rule — Multiplying  by  .1  moves  the  decimal  point  one  place  to  the  left;  multiply- 
ing by  . 01  moves  the  decimal  point  2 places  to  the  left.  (Taking  . 1 of  the  number  is 
really  dividing  by  10 — see  Rule  C,  p.  5). 


ARITHMETIC-PART  1 


1 


DIVISION 

Is  the  process  of  finding  how  often  one  number  is  contained  in  another. 

Divisor — is  the  number  by  which  we  divide. 

Dividend — is  the  number  divided. 

Quotient — is  the  answer  obtained  after  dividing. 

Remainder — is  the  number  left  over  from  the  dividend  if  the  divisor  is  not  con- 
tained in  the  dividend  an  exact  number  of  times.  The  remainder  is  a part  of  the 
dividend  and  should  always  be  less  than  the  divisor. 

The  Sign  of  division  is  -f-,  read  divided  by  and  means  that  the  number  before 
the  sign  is  to  be  divided  by  the  one  after  it ; as,  6 3 = 2. 

Division  is  also  shown  by  expressing  it  in  a fractional  form  the  dividend  with 

the  divisor  under  it  and  a straight  line  between ; 6 dividend  ^ 

— — ...  =2  quotient . 

3 divisor  2 

The  divisor  sometimes  is  put  in  front  of  the  dividend  with  a curved  line  between 
and  a straight  line  is  placed  over  the  dividend  to  separate  it  from  the  answer ; as, 
2 quotient 

divisor  3)6  dividend. 

When  the  dividend  and  the  divisor  are  like  numbers,  that  is  of  the  same  kind  or 
name,  the  quotient  is  abstract;  as  62  rds  -r-  32  rds.  = 2. 

Short  Division-  If  the  work  of  dividing  may  be  done  mentally,  the  divisor,  as  a 
rule,  not  exceeding  12,  only  the  answer  and  not  the  work  is  put  down,  as, 

1061 

6)637  or  6)687 


Beginning  at  the  left,  6 is  contained  in  600  one  hundred  times,  so  write  the  1 in 
hundreds’  place  or  directly  over  the  6;  6 is  contained  in  3 tens,  no  tens  times,  so  place 
a cipher  over  tens’  place.  We  then  have  still  remaining  the  3 tens,  which'are  equal  to 
30  units — these,  plus  the  7 units,  give  37  units — into  which  6 is  contained  6 times  and 
1 over,  so  place  the  figure  6 over  units  place  and  the  remainder  over  the  divisor  and 
add  it  to  the  whole  number ; as,  106£  Ans. 

In  actual  work  the  names  of  the  places  are  omitted  and  wc  simply  say  6 into  6 
once ; 6 into  3,  no  times  and  3 over ; 6 into  37,  six  times  and  1 over,  or  giving  alto- 
gether 106£. 

Proof  of  Division — Multiply  the  divisor  and  the  quotient,  and,  if  there  is  a re- 
mainder, add  it  to  the  product,  and  the  result  will  be  the  dividend. 

106  quotient 

X6  divisor 

636 

-(-1  remainder 

637  dividend 

Short  Methods  of  Division 

1.  To  divide  a whole  number  by  10,  take  off  one  place  at  the  right;  by  100,  two 
places,  and  the  numbers  not  cut  off  become  the  quotient,  and  the  figures  cut  off,  the 
remainder. 

100)63  | 72  gives  63  with  remainder  72 — expressed  63j7o20,  or  decimally  as  63.72. 

^ 2.  To  divide  a decimal  by  10,  move  the  decimal  point  one  place  to  the  left . by 
100,  two  places  to  the  left,  etc.  If  there  are  not  sufficient  figures,  prefix  ciphers, 
6.37  — r-  100  = .0637.  Answer—  .0637. 


8 


ARITHMETIC-PART  I 


Division 

21  into  6 no  times,  into  63 — three  times — place  in  the  quotient,  and 
multiply — 3 times  21  gives  63,  and,  subtracting,  we  have  no  remainder. 
Bring  down  the  next  figure,  7. — 21  into  7,  no  times,  placing  0 in  quo- 
tient; bringdown  the  next  figure  one,  1 — 21  into  71,  3 times,  place  the  3 
in  the  quotient — 3 X 21  = 63,  63  from  71  = 8,  the  remainder  or  28T. 
Ans.  303  ft . 

Division  of  Decimals 

2.54  + 

Always  make  the  divisor  a whole  number,  in  this  case  by  multiplying 
by  10.  In  order  not  to  change  the  quotient,  the  dividend  must  be 
multiplied  by  the  same  number  as  the  divisor  was.  Multiplying  by 
10,  move  the  decimal  point  1 place  to  the  right. 

Above  the  line  place  the  decimal  point  directly  over  the  decimal  point  in  the  dividend 
and  proceed  to  divide  the  same  as  in  whole  numbers.  If  there  is  a remainder,  and 
especially  if  there  are  several  decimal  places,  a + is  put  at  the  end  instead  of  the 
remainder. 

We  find  the  decimal  places  in  the  answer  are  equal  to  the  difference  in  the  number 
of  decimal  places  in  the  dividend  and  the  divisor;  as,  in  the  dividend  6.371  there  are 
3 decimal  places  and  in  the  divisor  2.5  there  is  one  decimal  place,  making  a difference 
of  2 decimal  places  - the  same  number  found  in  the  answer. 

The  figure  in  the  quotient  or  answer  each  time  must  be  placed  directly  over  the 
last  figure  brought  down  in  the  partial  dividend. 

Principles  of  Division: 

Multiplying  or  dividing  both  dividend  and  divisor  by  the  same  number  does  not 
change  the  quotient,  as 

50  -4-  10  = 5 

(50  X?)h-  (10  X 2)=  100  --  20  = 5 
(50  2)  -s-  (10  H-  2)=  25  -f-  5 = 5 

Multiplying  the  dividend  multiplies  the  quotient  by  the  same  amount,  as 
50  -r-  10  = 5 

(50  X 2)  -s-  10  = 100  10  = 10  or  5 X 2 

Multiplying  the  divisor  divides  the  quotient  by  the  same  amount. 

50  -5-  10  = 5 

50  (10  X 2)  = 50  20  = 2|  or  5 2 

Dividing  the  dividend  divides  the  quotient  by  the  same  amount. 

50  -s-  10  = 5 

(50  -r-  2)  -r-  10  = 25  -r-  10  = 2|  or  5 2 

Dividing  the  divisor  multiplies  the  quotient  by  the  same  amount. 

50  — 10  = 5 

50  -v-  (10  2)  = 10  or  5 X 2 


2.5.)  6.8.71 
W _50_ 
137 
125 
121 
100 
21 
25 


303  ft 
21)6371 
63 

71 

63 

_8 

21 


ARITHMETIC- PART  I 


9 


USE  OF  SIGNS  IN  COMBINATION 

Rules: 

1.  Use  -4-  and  X in  the  order  they  occur  from  left  to  right. 

2.  Use  + and  — in  the  order  they  occur  from  left  to  right. 

3.  If  all  4 signs  occur  -f-  X H in'  anY  combination  apply  1st  Rule  until  all 

and  X are  exhausted,  then  apply  Rule  2 until  all  -f-  and  — signs  are  exhausted. 


8 + 10—  6-3X7=? 
8 + 10  — 2 X? 

8 + 10  — 14 

18  — 14 


4 

4.  If  “ of”  occurs  in  an  expression  it  indicates  a closer  relation  than  a sign,  so 
perform  the  multiplication  indicated  by  the  “ of  ” first,  then  apply  Rule  1 and  then 
Rule  2 if  necessary. 

Parenthesis  (4  + 2)  X 7-  This  means  that  the  operation  indicated  within  the 
parenthesis  is  to  be  performed  first  and  regarded  as  one  whole ; so  6 X 7 = 42. 

Divisibility  of  Numbers: 

Figures,  or  any  of  the  ten  Arabic  characters  making  up  a number  are  called 
digits ; as,  in  23  two  and  three  are  each  a digit. 

A number  is  divisible  by 

2 —  if  the  units  digit  can  be  divided  by  2 or  0. 

3 —  if  the  sum  of  the  digits  can  be  divided  by  3. 

4—  if  the  two  right  hand  digits  are  0,  or  if  they  can  be  divided  by  4. 

5 —  if  units  digit  is  0 or  5. 

9— if  the  sum  of  the  digits  can  be  divided  by  9. 

FACTORING 


If  the  same  number  is  contained  an  exact  number  of  times  in  two  or  more  numbers, 
that  number  is  a common  divisor  of  all  the  numbers. 

If  it  is  the  greatest  number  that  will  be  contained  exactly  in  all  the  numbers,  it  is 
the  Greatest  Common  Divisor  (G.  C.  D.) 

Prime  Number  - is  one  whose  factors  are  1 and  itself. 

Composite  Number— is  one  that  has  other  factors  than  one  and  itself. 

Even  Number— is  one  divisible  by  2. 

Odd  Number— is  one  not  divisible  by  2. 

Factoring —Separating  numbers  into  factors. 


Method 

2 |_144 
3 |_72 
3 |_24 
2 | 8 
2J  4 
2 

Ans.  24  X 32 


Divide  the  number  by  one  prime  factor  and  the 
quotient  will  be  the  other  — divide  this  in  turn  by  a 
prime  factor  and  so  on  till  the  division  is  finished. 
All  the  divisors  and  the  last  quotient  are  the  prime 
factors,  for  if  they  are  multiplied  together  they  will 
produce  the  number. 

Exponent — The  little  figure  4 and  2 written  above 
and  to  the  right  of  the  factors,  indicate  how  many 
times  those  factors  have  been  used. 


10 


ARITHMETIC-PART  I 


Greatest  Common  Divisor: 

Find  the  G.  C.  D.  of  96  and  120. 

Method — Factoring  96  we  have  25  X 3. 

Factoring  120  we  have  2 3 X 3 X 5. 

So  the  Common  Divisors  of  both  are  3 and  23,  or  (3  X 2 X 2 X 2)  = 24,  the  G.  C.  D. 

Rule — Find  the  factors  common  to  both  numbers  and  multiply  them  together  for 
the  G.  C.  D. 

Old  Method  by  Long  Division  is  often  used  if  the  numbers  are  large. 

1 

96)120 

96  4 

G.  C.  D.  24)  96 
96 


Rule: 

Divide  the  greater  number  by  the  less,  then  the  last  divisor  by  the  last  remainder 
until  none  remains.  If  there  are  three  numbers  take  the  last  divisor  and  divide  it  into 
the  third  number,  continuing  as  before — the  last  divisor  being  the  Greatest  Common 
Divisor  (G.  C.  D.). 

The  principle  involved  being  that  a factor  of  two  numbers  is  also  a factor  of  their 
sum  and  difference. 

Least  Common  Multiple: 

Multiple— A number  containing  another  is  a multiple  of  the  number. 

Common  Multiple  — Is  a number  containing  each  of  two  or  more  numbers. 

Least  Common  Multiple  (L.  C.  M.)  of  two  or  more  numbers  is  the  least  number 
that  contains  each  of  the  two  or  more  numbers. 

Method — Least  Common  Multiple: 

I.  Find  the  prime  factors  of  each  number  given.  The  product  of  the  highest 
power  of  each  factor  (or  the  greatest  number-of  times  it  is  used  as  a factor  in  any  one 
number)  will  be  the  L.  C.  M. 

Example— What  is  the  L.  C.  M.  of  28,  21  and  15. 

28  = 2X2X7 

21  = 7X3  (L.  C.  M.  contains  2 twice,  7 once,  3 once, 

15  = 8X5  and  5 once.) 

2X2X7X3X  5 = 420  L.  C.  M. 

II.  Write  the  numbers  in  a horizontal  line  and  divide  by  any  prime  factor  which 
will  exactly  divide  two  or  more  of  the  given  numbers.  Directly  under  the  numbers 
copy  any  undivided  numbers  and  the  quotients  of  the  others. 

Use  these  as  a new  set  of  numbers  and  divide  as  before  until  nothing  but  the 
prime  factors  remain ; the  product  of  all  the  divisors,  and  of  the  numbers  on  the  last 
line,  will  give  the  L.  C.  M. 

Example — Find  the  L.  C.  M.  of  28,  21  and  15. 

3)28  21  15 

7)28  7 5 3 X 7 X 4 X 1 X 5 = 420  L.  C.  M.  Answer. 

4 1 5 

If  the  different  numbers  have  no  common  factor  the  Least  Common  Multiple  will 
be  the  product  of  the  different  numbers. 

CANCELLATION 

Dividing  both  dividend  and  divisor  by  the  same  number  does  not  alter  the  quotient, 


ARITHMETIC— PART  I 


11 


so  if  the  equal  factors  in  both  dividend  and  divisor  be  removed  it  shortens  the  work, 
and  this  is  known  as  Cancellation. 

4 

3 3 10 

W X U X W 36 

MX  0 ~ i ~ 

i 9 

1 l 


The  factor  5 is  canceled  or  removed  from  80,  leaving  16,  and  from  45  leaving  9 ; 
9 is  canceled  from  27  leaving  3,  and  from  9 leaving  1 ; 8 is  canceled  from  24  leaving 
3,  and  from  32  leaving  4;  4 is  canceled  from  16  leaving  4,  and  from  4 leaving  1.  The 
numbers  remaining  in  the  dividend  are  prime  to  those  remaining  in  the  divisor,  so  we 
can  cancel  no  further.  Then  perform  the  operation  indicated  by  the  signs  and  we  have 

86 

3 X 3 X 4 = 36  for  the  dividend,  and  the  divisor  IX  l = l,°r—  — 86.  Answer. 

FRACTIONS 

Fractions  are  the  equal  parts  into  which  a unit  is  divided. 

There  are  two  parts  to  each  fraction,  called  its  terms.  They  are: 

1.  Denominator  which  is  the  number  written  below  the  line  and  shows  into  how 
many  parts  the  unit  has  been  divided,  by  naming  them,  as  in  f,  four  shows  that  the 
unit  has  been  divided  into  four  equal  parts. 

2;  Numerator  which  is  the  number  written  above  the  line  and  shows  the  number  of 
those  parts  that  have  been  used;  as  in  three  gives  the  number  of  fourths  taken. 

Reading  a Fraction — Give  the  numerator  or  number  of  parts  first,  then  the 
denominator  or  name  of  the  parts;  as  f,  three-fourths. 

Proper  Fraction — One  whose  numerator  is  less  than  its  denominator,  indicating 
that  it  is  a portion  of  the  unit ; as  |. 

Improper  Fraction— One  whose  denominator  is  less  than  the  numerator,  making 
its  value  greater  than  a unit,  hence  is  not  properly  spoken  of,  as  part  of  a unit;  f = 1*. 

Mixed  Number— Is  a whole  number  and  a fraction;  as  2f,  read  2 and  three-fourths. 

Decimal  Fraction  is  one  whose  denominator  is  not  put  down,  but  is  10  or  a 
power  of  10.  It  begins  with  a decimal  point. 

Principles  of  Fractions: 

I.  Multiplying  the  numerator  or  dividing  the  denominator  increases  the  value  of 
the  fraction. 

4^2  g The  denominator  12  indicates  that  the  unit  has  been  divided  into 

a-  ^lS-  12  equal  parts.  The  numerator  4 indicates  that  4 of  the  12  parts 

have  been  taken.  Multiplying  the  numerator  by  2 gives  8 ; the 
parts  remaining  the  same  size,  we  have  increased  the  value  of  the  fraction  by  multi- 
plying the  numerator. 

4 4 The  denominator  12  indicates  that  the  unit  has  been  divided  into 

b-  12  -4-  2 “ 6 12  equal  parts,  and  the  numerator  4 indicates  that  4 of  the  ]2 

equal  parts  have  been  taken.  Dividing  12  by  2 gives  6,  but  if 
the  unit  has  only  been  divided  into  6 parts  instead  of  12,  each  part  must  be  twice  as 
large,  and  taking  the  same  number  of  parts,  that  in  this  case  are  larger,  the  value  of 
the  fraction  must  have  increased. 

II.  Dividing  the  numerator  or  multiplying  the  denominator  of  a fraction  decreases 
the  value  of  the  fraction. 

4 2 2 We  have  the  unit  divided  into  12  equal  parts  and  4 of  these  taken  ; 

a*  12  but  dividing  the  4 by  2 we  have  only  2 parts  taken  instead  of  4, 

and  as  the  size  of  the  parts  are  the  same,  the  value  has  been 
decreased. 


12 


ARITHMETIC-PART  I 


4 4 Twelve  indicates  the  number  ol'  equal  parts  into  which  the  unit 

k*  lax  2 =24_  has  been  divided;  multiplying  by  2 gives  24.  If  the  unit  has 
been  divided  into  24  parts  instead  of  12,  each  part  can  only  be 
half  the  size,  and  taking  4,  the  same  number  of  them,  the  value  of  the  fraction  has  been 
decreased. 

III.  Multiplying  or  dividing  both  terms  of  a fraction  by  the  same  number  does  not 
change  the  value. 

4^2  8 Denominator  12  indicates  that  the  unit  has  been  divided  into  12 

a"  12  X % ~2+  equal  parts,  and  the  numerator  that  4 of  those  parts  have  been 

taken.  Multiplying  each  term  by  2 gives  28?.  The  denominator 
indicates  that  the  unit  has  been  divided  into  24  equal  parts  instead  of  12,  so  each  one 
must  be  just  half  the  size.  If  8 parts  are  taken  instead  of  4,  we  have  twice  as  many 
parts,  but  each  part  is  half  the  size,  so  we  have  not  changed  the  value. 

4 g 2 Instead  of  12  equal  parts  the  unit  has  been  divided  into  6 equal 

12  -i-  2 ~ 6 parts,  so  each  part  must  be  twice  as  large,  but  we  have  taken 

only  half  as  many,  2 instead  of  4,  so  the  value  has  not  been 

changed. 

IV.  Adding  the  same  number  to  both  terms  of  a fraction  increases  its  value. 

412  6 Instead  of  12  equal  parts  the  unit  has  been  divided  into  14  equal 

parts,  so  the  size  of  the  parts  must  be  smaller,  but  instead  of  4 
parts  being  taken,  G of  the  smaller  parts  have  been  taken.  Sub- 


b. 


12  + 2 


14 


tracting  we  find  that  is 


2 ^ 4 

“2i  greater  than 

84 

6 


14 

4 

12 


36 


28 


8 2 

84  ~~  21 

V.  Subtracting  the  same  number  from  both  terms  of  a fraction  decreases  its  vatue. 

4 2 2 The  un^  has  been  divided  into  10  equal  parts  instead  of  12,  so 

42 2 the  parts  must  be  larger  ; but  instead  of  4 only  2 of  the 

larger  parts  have  been  taken.  Subtracting  we  find  that 
2 2 , , 4 

IT  is  IT  less  than  12 

60 

4 


20 


12 


60 


_2_ 

15 


REDUCTION  OF  FRACTIONS 

This  is  changing  the  form,  but  not  the  value  of  the  fraction. 

I.  To  change  whole  or  mixed  numbers  to  fractions  ; as  2|  to  4ths.  In  one  unit 
there  are  4 fourths,  so  in  2 there  will  be  2 X 4 or  8 fourths ; this  plus  the  3 fourths 
given  make  y. 


ARITHMETIC— PART  I 


13 


Rule — Multiply  the  whole  number  by  the  denominator,  add  the  numerator  and 
write  the  whole  result  over  the  denominator. 

II.  To  change  fractions  to  whole  or  mixed  numbers;  as  In  one  there  are  4 
fourths  (|),  so  in  15  there  will  be  as  many  as  4 is  contained  times  in  15,  or  3 and  3 
fourths  over  = 31- 

Rule— Divide  the  numerator  by  the  denominator  and  the  answer  equals  the  whole 
or  mixed  number. 


III.  To  reduce  to  higher  or  lower  terms. 

Principle— Multiplying  or  dividing  both  numerator  and  denominator  of  a fraction 
by  the  same  number  does  not  change  its  value. 

a.  To  reduce  to  higher  terms  multiply  both  terms  of  the  fraction  by  the 
number  which  gives  the  desired  denominator ; as  f to  44ths:  4 is  contained  in  44  eleven 
times — multiply  3 by  11  and  4 by  11,  thus  obtaining  af.  Answer. 

b.  To  reduce  to  lower  terms — divide  both  terms  of  the  fraction  by  the  same 

number. 

75  -s-  3 25  + 5 5 

105  -j-  3 ~ 35  h-  5 “ 7 

We  find  that  both  terms  are  divisible  by  3 and  5,  and  dividing  by  these  we  have  f . 
As  both  of  these,  5 and  7,  are  prime  to  each  other,  it  is  in  its  lowest  terms. 

In  reducing  to  lowest  terms  one  may  also  divide  both  terms  directly  by  their 

„ ^ ^ 75  15  5 

G.  C.  D„  as  Jog -35  =~7~- 


IV.  To  reduce  common  and  decimal  fractions.  Fractions  are  called  decimals 
if  they  express  a number  of  tenths,  hundredths,  etc.,  and  are  written  .61,  the  denomi- 
nator being  omitted  and  the  decimal  point  used  instead. 

To  express  decimally:  Place  a decimal  point,  then  write  the  number  and  see 
that  there  are  as  many  places  as  there  are  ciphers  in  the  denominator.  If  there  are 
not  a sufficient  number  of  figures,  prefix  ciphers  to  them;  as,  i0500 — decimal  point — 
3 ciphers  in  the  denominator  indicates  3 decimal  places;  but  there  is  only  one 
figure  given,  5,  so  prefix  2 ciphers,  giving  .005,  so  that  five  stands  in  the  place  called 
for  by  the  denominator  (thousandths). 

a.  To  reduce  decimals  to  fractions — .025  expressed  as  a fraction  with  a denomi- 
nator, f§00,  and  reducing  to  lowest  terms,  gives 

b.  To  reduce  fractions  to  decimals  ; as,  |.  The  value  of  the  fraction  is  found  by 
dividing  the  numerator  by  the  denominator. 


Divide  5 by  8,  annexing  decimal  ciphers  to  5,  gives 
. decimals. 


.625 

8)5.000 


as  in  the  division  of 


i of  5 is  the  same  as  saying  | of  50  tenths,  which  is  six  tenths,  and  the  2 tenths 
remaining  is  equal  to  2 hundredths,  dividing  that  by  8,  etc. 

V.  To  reduce  fractions  to  the  Least  Common  Denominator. 

Fractions  with  the  same  denominators  have  Common  Denominators. 

If  the  Common  Denominator  is  the  L.  C.  M.  of  the  given  denominators,  we  have 
the  Least  Common  Denominator.  (L.  C.  D.) 

If  this  L.  C.  D.  cannot  be  determined  by  inspection,  factor  the  denominators, 
multiply  all  prime  factors  the  greatest  number  of  times  they  occur  in  one  factor,  and 
the  result  is  the  L.  C.  D. 

Reduce  to  L.  C.  D.  f,  |,  J. 

The  L.  C M.  of  the  denominators  4,  8,  6,  is  24-  Making  24  the  denominator  of 


14 


ARITHMETIC— PART  I 


each  fraction,  we  find  4 into  24  goes  6 times ; multiplying  both  terms  by  6,  we  have 
3 X 6 = 18 
4X6  24 

5 X 3 = 15 


8X3  24 

5 X 4 = 20 
6X4  24 


and  8 into  24  goes  3 times ; multiplying  both  terms  by  3,  we  have 

and  6 into  24  goes  4 times;  multiplying  both  terms  by  4,  we  have 
4 8 15  2 0 Ans 

2 4’  24*  24 

Addition  and  Subtraction  of  Fractions 

Rule — To  add  or  subtract  fractions,  they  must  be  of  the  same  kind  or  similar 
fractions ; that  is,  having  the  same  or  a common  denominator. 

12 

g Fractions  not  being  similar  must  be  reduced  to  the  same 

3 denominator. 


5§ 

2% 


9i  i n = ii92 

L.  C.  D.  12  is  written,  as  indicated,  above  the  column.  3 into  12,  4 times ; multi, 
plying  both  terms,  gives  , but  writing  the  12  above,  will  avoid  its  repetition 
for  each  fraction ; 4 into  12,  3 times,  multiplying  both  terms,  gives  j* 2 3 4 * *2  ; 6 into  12, 
twice,  multiplying  both  terms,  gives  8 twelfths  -f-  3 twelfths  -j-  10  twelfths  gives 
||,  which,  being  reduced,  gives  1 whole  number  and  Y92.  The  fraction  Y92  can  be  re- 
duced to  the  lowest  terms  by  dividing  both  terms  of  the  fraction  by  3,  giving  f. 
Add  the  whole  number  one  to  the  other  integers,  giving  in  all  9|,  Answer. 

Reducing  to  a Common  Denominator,  we  have  8,  which 
8 gives  us  respectively  i and  |,  but  | cannot  be  taken  from 

54  HT+  $ = 9 e<  so  take  1 from  the  integer  6,  leaving  that  5;  the  1 taken 


being  equivalent  to  § ; this  -f- 


4 I 


gives  | 
1 from 


and 
5 is  4. 


; from 
Giving 


and  as 
or  sub- 


k-ns*  4 % | gives  |.  Subtract  the  integers, 

for  the  answer,  4|. 

In  examples  where  the  fractional  parts  are  expressed  both  as  decimals 
common  fractions,  change  the  common  fractions  to  decimals  before  adding 
tracting. 

Multiplication  of  Fractions 

I.  Multiplying  a fraction  by  an  integer. 

Multiplying  the  numerator  or  dividing  the  denominator,  multiplies  the  fraction. 
1X3=  = 2|  or  2 1 

| 3 = | or  2|  This  may  be  simplified  by  expressing  the 


5 1 5 

multiplication  and  canceling—  X">j  =~2  : 

2 


2 


i 

2* 


II.  Multiplying  a fraction  by  a fraction. 

X sign  is  read  of.  This  gives  a compound  fraction. 

| X I = ? | multiplied  by  1 gives  §,  so  § X I equals  | of  § or  2\,  and  § X I will 
be  seven  times  225  or  ||  reduced,  giving  T72. 

? 7 7 

Simplified  by  cancellation  -g  X"^~  gives 

4 

Ru|e — Multiply  the  numerators  together  for  the  product  and  the  denominators 

together  for  the  product. 


ARITHMETIC— PART  I 


15 


When  multiplying  by  a fraction,  we  multiply  by  the  numerator  and  divide  by 
the  denominator,  but  the  order  makes  no  difference,  as  § of  21.  We  may  say  2 times 
i of  21,  which  would  be  two  times  7,  or  14.  In  the  example,  § of  11,  it  is  easier  to  say 
t of  2 X H>  f°r  3 is  not  exactly  contained  in  11 ; so  we  have  £ of  22,  or  7£. 

III.  Multiplying  an  integer  or  mixed  number  by  a fraction. 

a.  Express  mixed  numbers  as  fractions  before  beginning  and  whole  numbers  as 
fractions  by  making  1 the  denominator. 

1 

11  4 2 22 

2|  X 4 X I =~J~X  Y X 3“  = 3^  = 73-  Answer. 

1 

Cancel  where  possible,  multiplying  the  numerators  for  the  product  and  the  de- 
nominators for  the  product.  If  the  answer  is  an  improper  fraction,  change  it  to  a 
mixed  number.  If  the  answer  is  a fraction  not  in  the  lowest  terms,  it  shows  that  all 
cancellation  has  not  been  done. 

b.  In  multiplying  a whole  number  by  a mixed  number,  it  is  sometimes  simpler 
to  use  this  method. 

215  multiplied  by  |,  as  above,  it  would  be  \ of  215  X 3 ; so 
multiplying  215  by  3 gives  645  and  dividing  by  4 gives  1611, 
then  multiply  the  whole  numbers  as  usual,  placing  each  partial 
product  under  the  figure  being  used  as  the  multiplier,  and 
uniting  this  result  with  the  result  of  multiplying  by  f , gives 
79011  for  the  answer. 

DIVISION  OF  FRACTIONS 

I.  To  divide  a fraction  by  an  integer. 

Rule— Divide  the  numerator  or  multiply  the  denominator  of  the  fraction  by  the 
integer. 

4 4-^-2  2 (used  if  the  numerator  contains  the  divisor  an  exact 

5 5 5 number  of  times). 

7 7 7 (used  if  the  numerator  does  not  contain  the  divisor 

9 ' 9X2  18  exactly). 

Or  by  cancellation: 

7 

21  n X 1 7 

5 ^9—  5 x ? ” ' 15 
3 


215 

36| 

4)645 

1611 

1290 

645 

79011 


II.  To  divide  either  a fraction  or  an  integer  by  a fraction. 

Rule — Change  the  denominators  to  a Common  Denominator  and  divide  the 
numerators ; or  invert  the  divisor  and  cancel. 


11)  128 
ll-/TAns. 


a 5 5 15  20  15  3 

* 8 6 = n ^ U “ 20  “ 4 

b.  _5  _ _L_JLV  8 __3_ 

8 * 6 $ S'  ft  4 

4 

The  (a)  method  is  sometimes  used  when  a mixed  number  is  to  be 
divided,  as  in  example  given,  changing  both  to  4ths,  and  dividing, 


so  that  128  fourths  contains  11  fourths,  11  andy-r  times. 


16 


ARITHMETIC— PART  I 


A complex  fraction  is  one  whose  numerator  and  denominator  are  either  or  both  a 
fraction. 


a. 


b. 


(1) 

(2) 


? Simplify  the  numerator,  then  the  denominator,  and  divide 
the  results  (numerator  by  denominator). 


£ 

6 


7_ 

6 


3 

_L  _?LV 

6 — 3?  A 

16 


3 

0 9 A 

~Y~  = jg-  Answer. 

1 


Fractional  Relations  and  Solutions 

I.  To  find  what  part  one  number  is  of  another. 

a.  What  part  of  3 is  2?  Since  1 is  £ of  3 ; 2 is  2 times  £ of  3 or  § of  3,  that  is 
2 is  § of  3. 

b.  What  part  of  f is  §?  Since  f = and  £ = T42,  then  £ is  the  same  part  of 
| that  r«2  is  of  t92  ; but  that  is  the  same  part  as  the  numerator  9 is  of  the  numerator  4, 
or  2\.  The  same  answer  is  obtained  by  dividing  f by  §,  which  = | X f = |.  or  2\. 

II.  To  find  the  whole  when  a pa.rt  is  given. 

a.  If  | of  a number  is  20,  what  is  the  number?  If  20  is  2 fifths,  one  will  be 
\ of  20  or  10,  and  the  whole  § will  be  5 times  as  much  or  50. 

b.  § is  f of  what  number?  Since  § is  f of  a number,  \ would  be  \ of  f , or  |. 
If  i is  £ the  whole  f = 7 times  £ or  ^ or  3£.  Answer. 


Analysis 

Reasoning  from  something  given  to  o?ie , and  from  one  to  the  required  quantity. 

1.  If  it  takes  7 boxes  to  fill  1 case,  how  many  boxes  are  needed  for  5 cases? 
Solution — If  1 case  requires  7 boxes,  5 cases  will  require  5 times  as  many,  or  7 X 

5 = 35  boxes. 

2.  If  peaches  are  3 for  10  cents,  what  will  15  cost? 

Solution — If  3 cost  10  cents,  one  will  cost  £ of  10,  or  3£  cents.  If  one  costs  3J  cents, 

5 

10  W 

15  will  cost  15  times  that  amount,  or  X "j-  = 50  cents.  Answer. 

1 


3.  If  1 book  cost  20  cents,  how  many  books  can  be  bought  for  $2.00? 

Solution — If  one  cost  $.20  we  can  buy  as  many  as  $.20  is  contained  m $2.00,  or  10. 

4.  The  box  contains  75  blue  envelopes  and  25  white  envelopes,  what  part  is  blue 
envelopes? 

Solution — 75  plus  25  gives  the  entire  number,  or  100.  Out  of  the  100  we  have  75 
blue,  or  which  is  f of  the  box. 

5.  If  4|  lbs.  cost  80  cents,  what  will  7|  lbs.  cost? 

Solution — If  4|  lbs.  cost  80  cents,  one  pound  will  cost  as  much  as  4|  is  contained 
times  in  80,  or  $.80  h-  4£  = $.80  X I = — = $.17|. 


4 


ARITHMETIC— PART  I 


17 


If  one  pound  cost  17|  cents,  lbs.  will  cost  times  that  amount,  or  17|  X 7 J'= 

$ .80  5 

.160  16  4.00 

— p — X ~2~  — XT  = ^1-333-  Answer. 

3 1 

6.  If  t56  of  the  men  are  away  and  143  remain  in  the  village,  how  many  went 
away? 

Solution — If  represents  the  whole  number  of  men,  T5g  went  away,  which  left  (If  — 
Ty  = in  the  village,  so  {A  of  the  men  = 143  men.  If  that  is  11  sixteenths,  one  will 
be  of  143  or  13  men ; and  if  is  13,  the  whole  j-f  will  be  16  times  as  much,  or  16  X 
13  .=  208  men.  Answer. 

7.  A man  lost  £ of  his  money  and  spent  | of  what  remained.  He  then  had  $18. 
What  did  he  have  at  first? 

Solution — The  whole  of  his  money  was  § ; he  lost  so  must  have  had  § remain- 
ing. Of  this  he  spent  or  \ of  f , which  is  He  had  § and  spent  £,  so  he  must  have 
had  left  (f  — £),  or  £.  But  £ was  equal  to  $18,  so  the  whole  f must  have  been  3 times 
as  much  or  $54. 

8.  A can  do  a piece  of  work  in  4 days  and  B in  6 days.  How  long  will  it  take 
them  working  together? 

Solution — If  it  takes  A 4 days,  in  one  day  he  will  do  \ of  the  work. 

If  it  takes  B 6 days,  in  one  day  he  will  do  £ of  the^work. 

Both  working  together,  in  one  day  they  will  do  £ + £,  or  ^ • 

To  do  the  whole  work  if,  it  will  take  as  many  days  as  the  amount  done  in 
one  day,  is  contained  in  J | ; or, 

1 

12  5 12  12 

“i2  -*■  i2"  = wy' t = x = 25  days-  Answer- 

i 

9.  There  are  two  pipes  in  a cistern ; one  fills  it  in  40  minutes  and  the  other 
empties  it  in  60  minutes.  If  the  cistern  is  empty,  and  both  pipes  started  at  once,  how 
long  will  it  take  to  fill  the  cistern? 

Solution — Find  the  least  number  which  will  hold  each  of  them,  or  the  L.  C.  M.  of 
60  min.  and  40  min.,  which  is  120  min.  So  the  first  pipe,  if  it  fills  it  once  in  40  minutes, 
will  fill  it  three  times  in  120  minutes;  and  if  the  second  empties  it  once  in  60  minutes, 
will  empty  it  twice  in  120  minutes;  hencS  the  cistern  filled  3 times,  minus  2 times 
emptied  in  120  minutes,  will  be  filled  once  in  120  minutes,  or  2 hours,  both  pipes  work- 
ing together.  Once  in  2 hours.  Answer. 

10.  There  is  food  enough  to  last  2,000  men  for  30  days,  but  1,000  leave.  How 
long  did  the  food  last  the  others? 

Solution — If  it  lasted  2,000  men  30  days,  it  will  last  one  man  2,000  X 30,  or  60,000 
days. 

Removing  1,000  men  leaves  1,000.  If  it  lastsone  man  60,000days  it  will  last  1,000 
men  as  many  days  as  1,000  is  contained  in  60,000,  or  60  days. 

11.  A boat  will  go  down  the  river  36  miles  in  3 hours,  but  it  takes  4 hours  to  come 
back.  What  is  the  rate  of  the  current  of  the  river? 

Solution — If  it  goes  36  miles  in  3 hours,  in  1 hour  it  will  go  12  miles.  As  it  is  going 
down  stream  it  is  aided  by  the  current,  so 

Rate  in  still  water  -f-  rate  of  the  current  = 12  miles  in  1 hour,  but  returning  it 
takes  4 hours  to  go  36  miles,  or  9 miles  in  1 hour.  This  time  the  current  lessens  the 
speed,  so 


18 


ARITHMETIC— PART  I 


Rate  in  still  water  — rate  of  current  = 9 miles  per  hour.  So  the  current  must 
have  made  a difference  of  3 miles  in  1 hour,  but  this  current  was  met  both  ways — 

2 X rate  °f  the  current  = 3 miles  in  1 hour,  therefore  the  rate  of  the  current  must 
have  been  one-half  of  3,  or  1|  miles  an  hour.  Answer. 

12.  To  3 times  a number  add  56,  divide  the  sum  by  7,  subtract  6 from  the  quotient, 
and  the  remainder  will  be  20.  What  is  the  number? 

Solution — Beginning  at  the  end  we  have  the  remainder  20  and  the  subtrahend, 
the  number  subtracted  is  6,  hence  we  have  20  + 6 = 26,  the  minuend,  which  was  also 
the  quotient  of  the  part  before,  so  we  have  divisor  7,  quotient  26,  to  find  the  dividend 
or  26  X 7 = 182  dividend ; but  this  was  also  the  sum  of  the  previous  part,  56  had  been 
added  to  three  times  the  number  and  gave  182;  so  182  — 56  = 126,  which  is  3 times 
the  number  . •.  the  number  must  be  r of  126,  or  42.  Answer. 

13.  The  sum  of  two  numbers  is  56,  their  difference  is  28,  what  are  the  numbers? 


Solution  A. 


Greater  -f  less  = 56 
Greater  — less  = 28 


2 X greater  = 84 
greater  = **  or  42  ) Answer 
56  — 42  14  f Answer- 


Sum  -)-  Difference 

— „ — = greater  nnmber. 


14. 


Solution  B. 


Sum  — Difference 


2 

56  + 28 
2 


less  number. 


14  | 

J 


Answer. 


1.  At  what  time  between  1 and  2 will  the  hands  of  a clock  be  together? 

Solution — Two  hands  are  together  at  noon  12  o’clock.  In  the  next  hour  the  minute 

hand  goes  through  60  minute  spaces  while  the  hour  hand  goes  through  5 minute  spaces, 
the  minute  hand  having  gained  55  spaces  in  the  hour. 

The  minute  hand  is  now  at  12,  and  the  hour  hand  at  one,  so  to  be  together  the 
minute  hand  would  have  to  go  5 more  spaces,  or  60  altogether. 

If  55  is  the  number  of  spaces  covered  in  1 hour,  it  will  take  as  many  hours  to  go 
60  spaces  as  55  is  contained  in  60,  or  §§  = 1^T  hour  = 

1 hour,  5 minutes,  27T\  seconds  p.  m. 

2.  At  what  time  will  they  be  together  between  4 and  5 o’clock? 

Solution— From  noon  the  minute  hand  would  have  to  go  60  minutes  each  hour  to 
be  together,  so  in  4 hours  it  would  have  to  gain  240  minutes  in  order  to  be  together 
with  the  hour  hand.  If  the  gain  in  1 hour  is  55  minutes,  to  gain  240  it  will  take  as 
long  as  55  is  contained  in  240,  or  = 4 hours,  21  minutes,  4911 2*  seconds  p.  m. 


RATIO 

Ratio  is  the  relation  that  one  number  bears  to  another  of  the  same  denomination. 
The  colon  (:)  is  the  sign  of  ratio. 

Ratio  may  be  expressed  9 : 3,  or  as  a fraction  § ; or  by  division  9 -5-  3. 

A ratio  is  always  abstract ; as,  the  ratio  $9  : $3  is  3,  or  $3  : $9  is  r. 

Antecedent  is  the  name  of  the  first  number  of  the  ratio. 


ARITHMETIC— PART  I 


19 


Consequent  is  the  name  of  the  second  number  of  the  ratio. 
Rule — Antecedent  divided  by  Consequent  = Ratio. 


PROPORTION 

This  is  an  equality  between  two  ratios. 

The  sign  of  proportion  is  the  double  colon  ( : : ) or  equality  (=)  sign  between  the 
ratios ; as,  4 : 8 : : 7 : 14,  or  4 : 8 = 7 : 14. 

It  is  read  4 is  to  8 as  7 is  to  14. 


(I  or  £)  = (i74  or  £) 

Extremes  are  the  first  and  fourth  terms. 

Means  are  the  second  and  third  terms. 

Principle  : 

a.  The  product  of  the  means  equals  the  product  of  the  extremes ; as, 

4 : 8 : : 7 : 14 
4 X 14  = 8 X 7 
56  = 56 


b. 

c. 


Product  of  the  means  divided  by  one  extreme  will  give  the  other  extreme, 
Product  of  the  extremes  divided  by  one  mean  will  give  the  other  mean. 


4 : 8 : : 7 : ? 
(means)  8X7 
Extreme  4 


= 14  other  extreme. 


(Extremes)  4 


14 

X 14 


8 other  mean. 


one  mean  7 

Rule  for  Solving  Problems : 

1.  Make  the  answer  the  fourth  term. 

2.  Take  for  the  third  term  the  number  that  is  of  the  same  kind  as  the  answer. 

3.  Decide  whether  the  answer  is  to  be  greater  or  less  than  the  third  term. 

4.  Arrange  the  remaining  numbers  for  the  first  and  second  terms  so  that  they 
will  bear  the  same  relation  to  each  other,  as  the  third  and  fourth  do  to  each  other. 
(Greater  to  less  or  less  to  greater). 

5.  Apply  principle  b. 

Example— What  will  20  tons  of  hay  cost  if  7 tons  cost  $42? 

Less  : greater  : : less  : greater 


20 


$42 


6 

20  X 4? 
1 

1 


$120. 


The  answer,  the  number  of  $ the  cost 
of  the  hay,  is  made  the  fourth  term. 
The  third  term  is  $42,  also  the  cost  of  the 
hay;  20  tons  cost  more  than  7 tons,  so  the 
ratios  are  arranged  less  to  greater. 


The  remaining  terms  are  7 and  20  tons  to  be  similarly  arranged  for  first  and  second 
terms,  or  7 : 20 — less  to  greater,  giving  the  proportion  7 : 20  : : $42  : ? $ 


MONEY 

It  is  a measure  of  value,  and  is  either  in  coin  or  paper. 

UNITED  STATES  MONEY 

Is  the  currency  used  by  the  United  States,  adopted  by  Congress  in  1786.  It  is  the 
decimal  system — dimes,  cents,  and  mills  being  written  as  decimals. 

The  coins  are  gold,  silver,  nickel  and  bronze. 

Gold  coins  now  coined  are  the  double- eagle,  $20;  eagle,  $10;  half-eagle,  $5.  Gold 
can  legally  be  given  for  any  amount  in  settlement  of  a debt. 


20 


ARITHMETIC- PART  I 


Silver  coins  are  the  dollar,  one-half  dollar,  one-quarter  dollar,  and  dime.  Silver 
can  legally  be  given  for  amounts  not  more  than  $5-00  at  any  one  time. 

Gold  and  silver  coins  are  T90  pure  metal,  ^ alloy,  in  order  to  give  hardness  to  the 
coin  and  so  prevent  its  wearing  off  so  quickly. 

Nickel  coins  are  the  five  cent  pieces.  They  are  made  of  nickel  and  copper. 

Bronze  coins  are  the  one  cent  pieces.  They  are  95  parts  copper  and  5 parts  tin 
and  zinc. 

Bronze  and  nickel  coins  can  legally  be  given  only  for  amounts  not  more  than 
twenty-five  cents  in  one  payment. 

The  mill  is  never  coined. 

Paper  Money  = $1,  $2,  $5,  $10,  $20.  $50,  $100,  $500,  $1,000,  $5,000  and  $10,000 
dollars  are  issued  m bills  and  are  used  instead  of  coin.  These  bills  are  gold  and  silver 
certificates,  bank  bills,  and  Treasury  notes  of  United  States. 

Names : 

Mill— from  the  Latin  mille,  meaning  thousand,  is  the  one-thousandth  part  of  a 
dollar. 

Cent — from  the  Latin  centum,  meaning  a hundred,  is  the  one-hundredth  part  of  a 
dollar.  (It  was  proposed  by  Robert  Morris.) 

Dime — from  the  French  disme,  meaning  tenth,  is  the  one-tenth  part  of  a dollar. 

Dollar — from  Dale-town,  where  it  was  first  coined. 

Eagle— from  the  national  bird. 

Table 

10  mills  = 1 cent  10  dimes  = 1 dollar 

10  cents  = 1 dime  10  dollars  = 1 eagle 

The  dollar  is  the  unit  of  measure.  When  cents  are  less  than  10  a cipher  must  be 
put  in  tenths  place,  as  $5.01. 

The  mill  is  placed  in  thousandths  place,  being  Tol0Q  of  a dollar.  In  answers  for 
business  purposes  5 or  more  mills  are  counted  as  another  cent,  less  than  five  mills  are 
not  taken  account  of. 

In  documents  such  as  checks,  etc.,  write  cents  not  as  decimals,  but  as  common 
fractions. 


Aliquot  Parts  of  a Dollar 


$.10  = A 

$.60  = § 

$.62£  = | 

$.50  = | 

$.66|  = I 

• 20  = 1 

12|=  h 

,87|  = | 

.75  = | 

•161  = h 

•40  = | 

II 

§5 

.25  = * 

*h!W 

II 

th'M 

CO 

CO 

■m  = i 

Canadian  Money 

In  1858  the  currency  of  Canada  was  changed  from  that  of  England  to  * a decimal 
one  with  coins  of  practically  the  same  value  as  those  of  the  United  States. 

The  gold  coin,  however,  is  the  English  Sovereign  worth  $4.8665. 

Silver  coins  are  the  fifty  cent  piece,  the  twenty-five  cent  piece,  shilling  or  twenty 
cent  piece,  dime,  half  dime. 

The  copper  coin  is  the  cent. 

The  Table  is  the  same  as  for  the  United  States  money. 

English  Money 

The  early  Germans  who  came  to  trade  with  the  English  were  called  Easterlings , 
or  those  coming  from  the  East.  Their  money  was  called  by  the  same  name,  but  has 
since  been  changed  to  Sterling. 

The  unit  is  the  pound  or  sovereign  = $4.8665,  but  in  estimating  values  $5.00  is 

used. 


ARITHMETIC-PART  I 


21 


Table 

4 farthings  = 1 penny  (d)  (denarius  Latin. ) 

12  pence  = 1 shilling  (s)  (solidus  “ ) 

20  shillings  = 1 pound  (:)  (libra  “ ) 

The  pound  value  was  secured  by  weight — 240  pence  making  one  pound.  The  sign 
£ always  precedes  the  number  of  pounds ; as,  £2  6s. 

Gold  coins  are  the  sovereign  or  pound;  half  sovereign  (10s.);  guinea  (21s.),  this  is 
not  coined  now.  It  was  so  named  because  the  first  one  was  made  from  gold  obtained 
in  Guinea,  Africa.  Half  guinea  (10s.  6d.)  not  coined  now. 

Silver  coins  are  the  crown  = 5s.  (not  coined  now) ; half  crown  = 2s.  6d.  (not  coined 
now);  florin  = 2s.,  shilling;  six  pence;  four  pence;  three  pence. 

Copper  coins  are  the  penny  and  half  penny. 

Farthing  means  four  things — at  first  the  English  penny  was  cut  by  a cross  so 
deeply  that  it  could  be  easily  broken  into  four  parts.  They  are  not  coined  now  and 
are  given  usually  as  the  fraction  of  a penny. 

French  Money 

The  unit  of  measure  is  the  franc.  The  money  is  the  decimal  system. 

100  centimes  (c.)  = 1 franc  (fr.) 

The  same  names  and  values  are  applied  to  the  money  of  Switzerland  and 
Belgium. 

The  units  of  Spanish,  Italian  and  Grecian  money  have  the  same  value  as  the 
franc. 

1 franc  = $.193 — but  in  quick  estimates  20  cents  is  used. 

German  Money 

The  unit  of  measure  is  the  mark , value  $.2385,  but  in  quick  estimates  taken  as 
$.25.  This  is  the  decimal  system. 

100  pfennings  (pf.)  = mark  (M.) 

MEASURES 

Capacity 

Dry  Measure — used  for  dry  substances.  The  unit  of  measure  is  the  bushel. 

2 pints  (pt.) =1  quart. . . .qt. 

8 quarts =1  peck. . . .pk. 

4 pecks = 1 bushel,  .bu. 

2|  bushels = 1 barrel,  .bbl. 

1 bushel  = 2150.42  cubic  inches. 

1 bushel  = 11^  cubic  feet. 

Liquid  measure  used  for  liquids.  The  unit  of  measure  is  the  gallon. 

4 gills  (gi) = 1 pint pt. 

2 pints =1  quart qt. 

4 quarts = 1 gallon gal. 

31 1 gallons =1  barrel bbl.  (These  are  not  fixed 

63  gallons =1  hogshead . . hhds.  f rates  in  business. 

1 gallon = 231  cubic  inches. 

7 \ gallons = 1 cubic  foot. 

Time  Measure 

This  is  determined  by  the  rotation  and  revolution  of  the  earth.  The  unit  of 
measure  is  the  day. 


22 


ARITHMETIC-PART  I 


Rotation  on  its  axis = 1 day. 

Revolution  around  the  sun. . = 1 year. 


60  seconds  (sec.) 

. = 1 minute 

60  minutes . . 

. — 1 hour 

. hr. 

24  hours 

. — 1 day 

..da. 

365  days 

. — 1 common  year. 

.yr. 

366  days. 

. — 1 leap  year 

.yr. 

100  years 

. — 1 century 

..cent. 

7 days 

. — 1 week 

4 weeks 

13  lunar  months,  1 da.  6 hr.  ) 

. — 1 lunar  month. . 

I 

.mo. 

or 

12  calendar  months j 

- — 1 year 

.yr. 

Calendar 


Julius  Caesar  changed  the  calendar  in  46  B.  C.,  making  the  year  365^  days,  thus 
giving  a little  too  much,  so  that  by  1682  this  made  the  time  10  days  ahead  of  what  it 
should  be.  Pope  Gregory  dropped  10  days  so  that  the  Spring  Equinox  would  fall  on 
March  21st,  and  decided  that  only  centennial  years,  divisible  by  400,  should  be  leap 
years,  that  is  a year  having  the  extra  day  making  it  366  days  long.  Most  Catholic 
countries  made  the  change ; but  England  especially  bitter  against  the  Catholics  at  that 
time  did  not  make  the  change  till  1752,  the  mistake  then  being  11  days.  Russia  and 
Turkey  never  made  the  change,  and  so  are  now  about  12  days  behind  the  United 
States.  The  beginning  of  the  year  was  also  changed  to  January  1st  instead  of  March 
25th. 

The  true  year  is  the  time  reckoned  according  to  the  sun,  and  consists  of  365  days 
5 hours  48  minutes  49.7  seconds. 

To  have  a whole  number  of  days,  instead  of  using  fractions,  the  ordinary  year  is 
taken  as  365  days  and  the  leap  year  as  366  days. 


Leap  Years 

a.  Those  years  that  are  divisible  by  4,  except  the  years  ending  in  00. 

b.  Those  years  ending  in  00  that  are  divisible  by  400. 

The  year  now  has  12  months — 4 seasons,  each  3 months  long. 

January  — Janus,  god  of  the  year. 

February  — Febru,  Roman  festival  celebrated  on  the  15th. 

(The  above  months  were  added  later,  as  at  first  there  were  only  10  months  in  the  year.) 
March  — Mars,  god  of  war  (first  month  of  the  Roman  Calendar). 

April  — Latin  Operire — meaning  to  open — shown  in  the  growth  of  the  vegetable 

kingdom. 

May  — Maia — Mother  of  Mercury — sacred  to  her. 

June  — Jrmo — wife  of  Jupiter — sacred  to  her. 

July  — Birth  of  Julius  Csesar — the  month  was  first  called  Quintilis 
August  — Augustus  Csesar  became  consul  in  this  month,  it  was  first  called 
Sextilis,  or  the  6th  month. 

September — 7th  month  "| 

October  — 8th  month  I According  to  the  old  reckoning  of  ten  months  to  the 


November — 9th  month 


Y 


December — 10th  month  j 


year,  March  being  the  first  month, 


ARITHMETIC-PART  I 


23 


The  number  of  days  in  the  months  can  easily  bo  learned  by  following : 
“ Thirty  days  hath  September, 

April,  June,  and  November; 

All  the  rest  have  thirty-one, 

Excepting  February  alone; 

Which  hath* but  twenty-eight; 

Till  leap  year  gives  it  twenty-nine.” 


Paper  Measure 

24  sheets  = 1 quire. 

20  quires  ) 

or  r = 1 ream. 

480  sheets  ) 

Weight 

Measures  estimated  by  a scale  or  a balance. 


Counting 

12  things  = 1 dozen. 

12  dozen  = 1 gross. 

12  gross  = 1 great-gross. 
20  things  = 1 score. 


Troy  Weight  is  used  for  weighing  gold,  silver,  etc. 


24  grains  (gr.) = 1 pennyweight  (pwt.) 

20  pennyweights = 1 ounce  (oz.) 

12  ounces = 1 pound  (lb.) 

5760  grains =1  pound  Troy. 

3.168  troy  grains = 1 carat  (used  in  measuring  jewels). 


14_ 

24 


A carat  used  for  measuring  gold  means  of  the  whole,  so  14  carats  would  be 


gold,  the  rest  alloy. 


Avoirdupois  Weight  is  used  for  weighing  articles  other  than  gold,  silver,  etc. 


16  ounces  (oz. ) = 1 pound  (lb. ) 

100  pounds. =1  hundred-weight  (cwt.) 

2000  pounds  or  20  cwt =1  ton  (short) 

2240  pounds  =1  ton  (long).  Not  used  generally  except  in 

Custom  Houses. 

7000  grains = 1 pound  (avoirdupois). 

1 cubic  foot = 62 1 pounds. 

1 barrel = 196  pounds. 


It  is  used  to  measure  distance. 

12  inches  (in. ) 

3 feet 

16* feet  ) 


5 1 yards  ) 
320  rods  j 


5280  feet  ) 
4 inches . . 
1 inch 


Length 

= 1 foot  (ft.) 

= 1 yard  (yd.) 

= 1 rod  (rd.) 

= 1 mile  (mi.) 

= 1 hand  ( used  in  measuring  horses ). 
= 3 sizes  (used  in  measuring  shoes'). 


Surface — Square  Measure 

Square  measure  is  used  to  measure  surfaces  of  land,  streets,  rooms,  etc. 
144  square  inches  (sq.  in.)  = 1 square  foot  (sq.  ft.) 

9 square  feet  (sq.  ft.). . . = 1 square  yard  (sq.  yd.) 

30^  square  yards  ) 

or  >- = 1 square  rod  (sq.  rd. ) 

272^  square  feet  ) 

160  square  rods = 1 acre  (A) 

640  acres = 1 square  mile  (sq.  mi.) 

100  square  feet =1  square  (roof,  floors,  etc.) 


24 


ARITHMETIC -PART  I 


Volume — Cubic  Measure 

Cubic  measure  is  used  for  measuring  articles  having  three  dimensions,  length , 
breadth  and  thickness. 

1728  cubic  inches  (cu.  in.) . =1  cubic  foot  (cu.  ft.) 

27  cubic  feet = 1 cubic  yard  (cu.  yd.) 

2150.42  cubic  inches =1  bushel.  * 

1|  cubic  foot . = 1 bushel. 

231  cubic  inches =1  gallon. 

1 cubic  foot = 62 1 pounds. 

Wood 

Cord  of  Wood  is  so  called  because  the  pile  was  first  measured  by  a cord,  the 

long  high  wide 
dimensions  of  the  pile  being  g-^r  X 4“ ft  X 4“ ft" 

Cord  Foot  was  one  foot  in  length  of  the  above  pile. 

16  cubic  feet. =1  cord  foot  (cd.  ft.) 

128  cubic  feet  ) 

or  >- =1  cord  (cd.) 

8 cord  feet  ) 

Circular  Measure 

This  is  used  in  measuring  angles — latitude  and  longitude. 

60  seconds  (B) =1  minute  (') 

60  minutes = 1 degree  (p) 

360  degrees =1  circumference. 

The  circumference  of  a circle  contains  360°  whether  the  circle  is  large  or  small. 

Longitude  and  Time 

15°  of  longitude =1  hour  of  time. 

15'  of  longitude = 1 minute  of  time. 

15"  of  longitude =1  second  of  time. 

DENOMINATE  NUMBERS 
Reduction 

Reduction  Descending — changing  to  lower  denominations  without  changing  the 
values. 

Examples: 

1.  Change  5 gallons,  2 quarts,  1 pint— to  pints. 

5 gal. 

4 

— Method — There  are  4 quarts  in  1 gallon,  hence  in  5 gallons 

^2  ^ S there  are  5X4  or  20  quarts,  this  added  to  the  quarts 

— given  in  the  example  makes  22  quarts. 

^2  ^ In  1 quart  there  are  2 pints,  therefore  in  22  quarts  there  are 

— 22  X 2 or  44  pints,  this,  plus  the  1 pint  of  the  example,  gives 

44  \ 

l 45  pints. 

45  pts.  Answer. 

2.  Change  | miles  to  inches. 

7 40  140  11  J 

a.  p mi.  X rds.  = 280  rds.  b.  280  rds.  X 5J  = X 0 ~ 1^40  yds. 

1 X 

yds.  ft. 

c.  1540  X 3 = 4620  ft.  d.  4620  X 12  = 55440  inches.  Answer. 


ARITHMETIC— PART  I 


25 


Reduction  Ascending — Changing  to  higher  denominations  without  changing  the 
value. 

Example : 

1.  How  many  rods  in  214  feet? 

3 ) 214  ft. 

.A  71  yds.  1 ft. 

2 / X 2 Answer  12  rods,  5 yards,  1 foot. 

11)  142  half  yds. 

12  rds.  10  half  yds.  or  5 yds. 


2.  Reduce  3 cwt.  70  lbs.  to  tons. 

70  lbs.  -v-  100  = t7o%  or  ^ cwt.  This  plus  3 cwt.  = 3/0  cwt. 

3/o  cwt*  -s-  20  = i o X — 2Y0  tons-  Answer. 

Since  100  lbs.  make  1 cwt.,  1 lb.  is  cw£.,  so  70  lbs.  would  be  70  times  1 or  /Q°6 
cwt.  or  /o  cwt.  This,  added  to  the  3 given,  makes  3/g  cwt. 

20  cwt.  = 1 ton,  so  1 cwt.  would  be  ^ of  a ton,  and  3T70  would  be  3T70  X 20  — 2Y0 
tons.  Answer. 

Addition,  Subtraction,  Multiplication,  Division 

is  the  same  as  for  other  numbers,  except  that  care  must  be  taken  to  use  the  correct 
number  of  units  of  one  denomination  in  changing  to  the  next,  as  they  vary  so  much. 


Add. 

15  ft.  8 in. 
2 ft.  10  in. 

17  W 
1 6 

Ans.  18  ft.  6 in. 

Subtract. 

14  20 

X^ft.  $ in. 
2 ft.  10  in. 
Ans.  12  ft.  10  in. 


10  in.  plus  8 in.  = 18  in. 

15  ft.  plus  2 ft.  = 17  ft. 

but  12  inches  makes  a foot,  so  in  18  inches  there  are  1 
foot  6 inches,  which  gives  a total  sum  of  18  feet  6 inches. 

10  inches  cannot  be  taken  from  8 inches,  so  change  1 foot 
to  12  inches,  giving  14  feet,  20  inches,  then  subtracting  2 
feet  10  inches  we  have  12  feet  10  inches. 


Difference  of  Dates 

Represent  years,  months  and  days  by  figures  and  subtract  as  in  denominate  num- 
bers. Always  count  30  days  to  a month  in  subtracting  unless  the  exact  number  of 
days  is  called  for. 

Example — What  is  the  difference  between  July  1,  1907,  and  December  12,  1905? 

18 

6 6+12  31 

1907  — 7 X 

1905  12  12 

1 6 19 


Difference  in  Hours 

There  are  24  hours  to  a day,  divided  in  groups  of  12,  one  set  before  noon  A.  M. 
and  one  set  after,  P.  M.;  but  in  subtracting  do  not  separate  into  these  groups — for  the 
time  after  noon,  or  p.  m.,  count  up  to  the  24th  hour. 

Example — What  is  the  difference  between  5 p.  m.  and  11  a.  m.? 

Hrs. 

17 

11  5 p.  m.  is  the  17th  hour  of  the  day. 

6 hours.  Answer. 

Writing  Time— the  numbers  representing  the  hours,  minutes  and  seconds  are 
written  one  after  the  other  with  a colon  between,  as,  10:20  meaning  20  minutes  after 


26 


ARITHMETIC-PART  i 


10.  The  minutes  arc  always  added  to  the  hour  they  follow,  as  20  minutes  of  eleven 
would  be  written  10.40  and  read  “ten  forty,’'  as  the  11th  hour  has  not  been  reached 
to  use  as  a starting  point  from  which  to  reckon. 

Multiply 

12  ft.  8 in.  Multiplying  by  5 gives  60  feet  40  inches,  but  40  inches 

contains  3 feet  4 inches,  making  in  all  63  feet,  but  63  feet 

fifd  ft.  0 in. 

3 ft.  4 in.  is  equal  to  21  yards,  0 feet,  so  the  answer  is  21  yards, 

Ans.  21yds.  0 ft.  4 in.  0 feet,  4 inches. 

Division 


Divide  11  ft.  8 in.  by  4 4 into  11  feet  is  contained  twice  and  3 feet  over ; 3 feet  = . 

4)11  ft.  8 in.  36  inches,  this  plus  8 inches  = 44  inches;  this  divided  by 

2 ft.  11  in.  Ans.  4 = 11  inches. 

If  both  numbers  are  compound,  reduce  both  to  the  same  denomination  (the  lowest 
denomination  called  for)  and  then  divide. 

A vessel  contains  8 gallons  3 quarts,  how  many  of  the  smaller  vessels,  each  con- 
taining 3 quarts  1 pint,  will  it  take  to  fill  the  larger? 
a.  8 gal.  3 qts.  0 pts.  = 70  pts.  b. 

gal.  qts.  qts.  qts.  3 qts  1 pt.  = 7 pts. 

8 X 4 = 32  + 3 = 35  qts.  pt.  pt. 

35  qts.  X 2 = 70  pts.  3X2  = 6+  l==r7  pts. 

c.  70  pts.  — j—  7 pts.  = 10.  Answer. 

Note — In  denominate  numbers,  if  a fraction  occurs,  reduce  it  to  lower  denomina- 
tions and  add  it  to  the  proper  column. 

LONGITUDE  AND  TIME 

Longitude  is  the  distance  east  or  west  of  a given  meridian.  It  is  measured  in 
degrees,  minutes  and  seconds.  It  can  never  be  greater  than  180°. 

A Meridian  is  a half  of  a great  circle  passing  from  pole  to  pole.  It  is  sometimes 
called  the  mid-day  line,  for  all  places  on  that  line  have  mid-day  or  noon  at  the  same  time. 

The  Prime  Meridian  is  the  meridian  selected  to  measure  from. 

The  Meridian  going  through  Greenwich,  London,  is  generally  taken  as  the 
Prime  Meridian.  A place  on  this  meridian  has  no  longitude. 

Difference  in  Longitude — If  both  places  have  the  same  name,  i.  e.,  are  East  or 
both  West  subtract  to  find  the  difference,  but  if  they  have  different  names,  one  being 
East  and  the  other  West,  add  to  find  the  difference. 


A..  . 

. . 15°  E . 

C 

. . 10°  E. 

West 

East 

B 

5°  E. 

D . . . . 

. . 15°  W. 

5 

5°  15° 

s 

..... B . .A 

10°  Dif. 

25°  Dif. 

03 

j=3 

V ; 

15° 

10° 

D 

.c 

The  earth  revolves  on  its  axis  from  west  to  east  once  in  24  hours,  the  sun  appear- 
ing to  move  round  the  earth  from  east  to  west,  so  it  is  afternoon  for  places  east  of  the 
meridian  and  forenoon  for  places  west  of  it. 

Hence  places  East  receive  the  sun  first,  and  moving  on  will  have  later  time  than 
places  West,  just  receiving  the  sun. 

Therefore,  of  two  places,  no  matter  whether  both  are  West  Longitude  or  East 
Longitude,  the  easternmost  one  has  later  time  and  clocks  will  be  faster , while  the 
other  place  will  have  earlier  time  and  the  clocks  will  be  slower. 

In  24  hours  360°  pass  under  the  vertical  rays  of  the  sun. 

Rotation— 3609  in  24  hrs.  = 15°  in  1 hr 

Rotation— 15°  or  9001  in  60  min.  (1  hr.)  = 15'  in  1 min. 

Rotation — 15'  or  900"  in  60  sec.  (1  mm.)  = 15“  m 1 sec. 


Tables 

Difference  in  Longitude  = Difference  in  Time 

15°  (degree) =1  hour  of  time 

15'  (minute) = 1 minute  of  time 

15"  (second).  . =1  second  of  time 


Circular  Measure 
60  (")  seconds  = 1 minute  (') 

60  (')  minutes  = 1 degree  (°) 
360°  degrees  = 1 circumference 


ARITHMETIC— PART  1 


27 


Rules : 

1.  Difference  in  Time  = Difference  in  Longitude  15. 

2.  Difference  in  Longitude  = Difference  in  Time  X 15. 

Examples  : 

When  it  is  10  a.  m.  at  San  Francisco  122°  24'  32"  W.,  what  is  the  time  at  Washing- 
ton 77-  03'  06"  W.? 

Before  we  can  find  the  time  at  Washington  we  must  know  the  difference  in  time 
between  the  two  places,  so  use 

Rule  1.  Difference  in  Time  = Difference  in  Longitude  -s-  15. 

Before  applying  the  rule  we  must  know  the  difference  in  longitude. 

As  both  places  are  West  Longitude,  find  the  difference  by  subtracting  the  less 


from  the  greater. 


122r 

77c 


24'  32"  W. 
03'  06"  W. 


45°  21'  26"  Difference  in  Longitude. 

Applying  rule — Difference  in  Longitude  -t-  15  gives  difference  in  Time,  of  the 
degrees,  minutes,  and  seconds,  will  equal  the  “Difference  in  Time  ” in  hours,  minutes 
and  seconds. 


15  1 45°  21'  26" 

3 hrs.  1 min.  25J|  sec.  Dif.  in  Time 


West 
Wash.  77°. . . 

? 

S.  Frisco  122° 

10  A.M. 


East 


Places  East  have  later  time. 

We  see  that  though  Washington  is  West  Longitude  it  is  further  East  than  San 
Francisco,  so  must  have  later  time.  If  the  time  at  San  Francisco  is  10  a.  m.  and  the 
difference  in  time  is  3 hrs.  1 min.  sec. — at  Washington,  which  has  later  time,  it 

10  — 0 — 0 

must  be  the  13th  hour,  etc.,  or  1 : 1 : 25  f.  m.  3 — 1 — 

13  — 1 — 25H 

2.  What  is  the  longitude  of  Washington,  whose  time  is  1 hr.  1 min.  26  sec.  p.  m. 
when  it  is  10  a.  m.  at  San  Francisco,  whose  longitude  is  122°  24'  32"  West? 

Before  finding  longitude  of  Washington  we  must  know  the  difference  of  longitude. 
Rule — Difference  in  Time  X 15  = Difference  in  Longitude. 


hr. 

mm. 

sec. 

13 

1 

26 

10 

0 

0 

3 

1 

26 

Difference  in  Time. 

15 

45° 

15' 

m* 

Difference  in  Longitude. 

* 

+ 

390"  -f-  60  = 6'  — 30"  over. 

6 

30 

15'  plus  6'  = 21". 

21 

or 

21'  60  = 0°  — 21"  over. 

45° 

21' 

30" 

Difference  in  Longitude. 

122° 

45° 


24' 

21' 


32"  W. 
30"  W. 


77°  3'  2"  W. 


Wash.  1 p.  m. 

? ° 

(Dif.  45°) 

S.  Frisco  10  a.m 

122° 


Washington  having  later  time  must  be  further  east  and  so  nearer  the  prime 
meridian,  and  therefore  would  have  less  degrees,  so  we  subtract. 

International  Date  Line 

Persons  traveling  180°  west  would  see  the  sun  twelve  hours  after  they  would  see 
it  at  Greenwich.  Persons  traveling  180°  east  would  see  it  twelve  hours  before  they 
would  see  it  at  Greenwich. 

When  these  persons  meet  at  the  180th  meridian  from  the  place  of  starting,  they  will 
find  when  they  compare  reckoning  of  time  that  it  differs  in  a great  degree.  The  one 


2S 


Arithmetic-part  i 


who  has  gone  in  an  easterly  direction  will  find  that  his  time  is  twelve  hours  later 
than  when  he  started.  The  one  who  went  in  a westerly  direction  will  find  that  his  is 
twelve  hours  earlier  than  when  he  started.  Suppose  that  it  was  New  Year’s  Day  at 
noon  when  they  started,  the  one  who  went  east  would  find  that  by  his  reckoning  it 
was  twelve  midnight  of  January  first;  the  one  who  went  west  would  find  by  his  reckon- 
ing that  it  was  twelve  midnight  of  December  thirty-first — yet  they  are  together  at  the 
180th  meridian  from  where  they  started. 

To  overcome  this  difficulty  an  International  Date  Line,  at  the  180th  meridian  east 
and  west  of  Greenwich,  has  been  agreed  upon  by  the  nations  as  the  place  where  time 
is  changed.  Those  sailing  west  crossing  the  line  add  a day,  and  those  going  east 
crossing  the  line  subtract  a day  from  their  time. 

The  person  traveling  west  crossing  this  line  on  Sunday  would  by  this  agreement 
call  it  Monday.  The  persons  traveling  east  crossing  the  line . on  Sunday  would  call  it 
Saturday,  at  the  same  hour  of  the  day. 

This  line  does  not  exactly'  follow  the  180th  meridian,  but  avoids  land  bodies  as 
much  as  possible  and  so  is  irregular. 


Standard  Time 


For  convenience  of  railroads,  etc.,  the  United  States  has  been  divided  into  5 sec- 
tions, each  about  15°  (longitude)  wide.  All  places  within  a section  using  the  time  of 
the  central  meridian  of  that  section. 

The  time  of  each  district  is  the  solar  time  of  meridians  15°  apart. 

The  Districts  are  known  as : 

Atlantic  or  Colonial  - Solar  time  of  60  meridian  W.  L.  4 hours  earlier  than  Greenwich. 

Eastern  — Solar  time  of  75  meridian  W.  L.  5 hours  earlier  than  Greenwich. 

Central— Solar  time  of  90  meridian  W.  L.  6 hours  earlier  than  Greenwich. 

Mountain — Solar  time  of  105  meridian  W.  L.  7 hours  earlier  than  Greenwich. 

Pacific  - Solar  time  of  120  meridian  W.  L.  8 hours  earlier  than  Greenwich. 

As  the  railroads  change  the  time  of  running  only  at  important  stations  or  junc- 
tions, the  lines  of  division  of  the  districts  are  irregular. 

The  boundary  between  the  Colonial  and  Eastern — From  the  St.  Lawrence  River 
near  65  meridian,  west  longitude  irregularly  to  the  Atlantic  Coast  of  Maine,  near  the 
68  meridian  west  longitude. 

The  boundary  between  the  Eastern  and  Central  Districts — From  Fort  William  on 
the  Canadian  side  of  Lake  Superior,  it  follows  the  Canadian  lake  shore  line  to  Buffalo; 
thence  in  an  irregularly  southeasterly  direction  to  near  Gainesville,  Georgia;  thence 
southeasterly  to  the  coast  at  Savannah,  Georgia. 

The  boundary  between  the  Central  and  Mountain  Districts— From  a point  on  the 
northern  boundary  of  the  United  States  near  the  103  meridian  west  longitude,  south- 
easterly to  a place  in  Nebraska  near  Long  Pine ; thence  westerly  to  Alliance,  Nebraska ; 
thence  irregularly  in  a southwesterly  and  southeasterly  direction  to  Cheyenne;  thence 
easterly  to  the  Arkansas  River;  thence  southwesterly  to  the  boundary  of  the  United 
States  and  Mexico  near  El  Paso. 

The  boundary  between  the  Mountain  and  Pacific  Districts — From  a point  on  the 
boundary  of  the  United  States  and  Canada,  just  east  of  the  117  meridian  west  longi- 
tude, in  a generally  southwesterly  direction  to  a place  just  east  of  the  120  meridian 
west  longitude,  south  of  the  40°  parallel  north  latitude ; thence  in  a southeasterly 
direction  to  a point  near  El  Paso,  on  the  boundary  of  the  United  States  and  Mexico. 

There  being  15°  longitude  difference  between  each  section  gives  1 hour’s  difference 
in  time  in  each. 

Difference  in  Time  ==  Difference  in  Longitude  -4-  15. 

Solar  Time — Is  the  actual  time  of  a place.  It  differs  as  places  are  east  or  west  of 
a meridian  from  which  the  time  is  reckoned. 

Example  3.  When  was  a message  sent  from  Cairo  30°  East  to  London  0°  if 
received  there  at  5.15  p.  m.? 


30°  E.  West 

^ London  0° . 

15)30°  Difference  in  Longitude.  5pm 

2 hrs.  Difference  in  Time. 


East 

..Cairo  30° 

7 


Cairo  is  further  East,  so  must  have  later  time,  or  7.15  p.  m.  Answer. 

Example  4.  What  is  the  difference  between  the  standard  and  the  local  time,  if 
the  longitude  of  Boston  is  71°  3'  30"  E.? 


ARITHMETIC— PART  I 


29 


Boston  being  70°,  etc.,  must  be  in  the  section  using  75°  longitude. 

Difference  in  Time  = Difference  in  Longitude  -r-  15. 

59 

74  00  60 

0'  0"  E. 

J71Q_  3 30  E. 

15)  3V  56'  30"  Difference  in  Longitude. 

0 hrs.  15  min,  46  sec.  Difference  in  Time.  Ans. 

METRIC  SYSTEM 

It  was  invented  in  France  in  1800.  The  unit  of  measure  in  this  system  is  the 
meter.  It  is  nearly  one  ten-millionth  part  of  the  distance  between  the  equator  and  the 
poles;  or  nearly  39.37  inches. 

It  is  a decimal  system  and  in  each  table  there  is  joined  to  the  word  standing  for 
the  unit,  Latin  prefixes  to  indicate  the  decimal  parts  and  Greek  prefixes  to  indicate 
the  multiples. 

The  abbreviations  are  the  first  letter  of  the  prefix  and  the  first  letter  of  the  unit, 
both  written  with  small  letters  if  it  is  to  indicate  a part;  but  if  it  is  to  indicate  a 
multiple  the  first  letter  is  usually  a capital,  though  some  do  not  use  the  capital  at  all. 


Prefixes 

deci .....  ......  = .1  deka . . = 10 

cent.... = .01  hekto . =100 

mill ......  .001  kilo =1000 

Table  of  Length.  (Unit  = Meter) 

10  Millimeters  (mm.). = 1 centimeter  (cm.) 

10  Centimeters =1  decimeter  (dm.) 

10  Decimeters . . =1  meter  (m) 

10  Meters. . . =1  dekameter  (dm.) 

10  Dekameters.  . = 1 hektometer  (hm.) 

10  Hektometers.  =1  kilometer  (km.) 


The  above  measures  most  used  are  the  Centimeter  (scientific  work),  Meter 
(cloth),  Kilometer  (long  distance). 

The  table  is  on  the  scale  of  10,  so  in  reduction  one  decimal  place  must  be  allowed 
for  each  denomination;  as 

525.  m.  = 5250.  dm.  = 52500  cm.  or  525.  m.  = 52.5  Dm.  =5.25  Hm.  = .525  Km. 


Table  of  Capacity.  (Unit  = Liter) 

The  table  of  capacity  is  used  for  liquid  and  dry  measure. 

10  Milliliters  (ml. ) =1  centiliter 

10  Centiliters =1  deciliter 

10  Deciliters =1  liter 

10  Liters  ....  =1  dekaliter 

10  Dekaliters =1  hektoliter 


The  measures  most  used  are  the  Liter  (small  quantities)  and  the 
quantities). 


Table  of  Weight.  (Unit  = Gram) 

A gram  is  the  weight  of  1 cubic  centimeter  of  water. 

10  Milligrams  (mg.) = \ centigram 

10  Centigrams ...  = l decigram 

10  Decigrams  =1  gram 

10  Grams = 1 dekagram 

10  Dekagram . = 1 hektogram 

10  Hektogram =1  kilogram 

1000  Kilograms =1  metric  ton. 


(cl.) 

(dl.) 

(!•) 

(dl.) 

(HI.) 

Hektoliter  (larger 


(eg.) 
(dg.) 
(g-) 
(Dg.) 
(H  g.) 
(Kg) 


Table  of  Surface— Square  Measure.  (Unit  = Square  Meter) 

100  Square  millimeters  (sq.  mm.)  or  (mm2)  = 1 square  centimeter  (sq.  cm.)  or  (cm2) 


100  Square  centimeters = 1 square  decimeter  (sq.  dm.)  or  (dm2) 

100  Square  decimeters = 1 square  meter  (sq.  m.)  or  (m2) 

100  Square  meters.  =1  square  dekameter  (sq.  Dm.)  or  (Dm2)  = are 

100  Square  dekameters = 1 square  hektometer  (sq.  Hm.)  or  (Hm2)  = hektare 

100  Square  hektometers =1  square  kilometer  (sq.  Km.)  or  (Km2) 


30 


ARITHMETIC-PART  I 


The  above  measures  most  used  are  the  square  meter  (area  of  floors,  walls,  etc.), 
square  kilometer  (area  of  land,  etc.). 

The  names  are  (a)  and  hektare  ( ha ) are  used  in  measuring  land. 

This  table  increases  by  102  or  100,  so  in  reduction  two  decimal  places  must  be 
allowed  for  each  denomination;  as, 

52535  sq.  mm.  = 525  35  sq.  cm.  = 5.2535  sq.  dm.  = .052535  sq.  m.  = .00052535 
sq.  Dm.,  etc. 


Table  of  Volume— Cubic  Measure  (Unit  = Cubic  Meter) 

1000  cubic  millimeters  (cu.  mm.)  or  (mm3)  = 1 cubic  centimeter  (cu.  cm.)  or  (cm3) 
1000  cubic  centimeters =1  cubic  decimeter  (cu.  dm. ) or  dm3) 

1000  cubic  decimeters  cubic  met“  <cu'  m>  or  <m3> 


This  table  increased  by  103  or  1000, 
allowed  for  each  denomination ; as, 

2535  cu.  mm.  = 52.535  cu.  cm. 


1 stere  (wood  measure) 
so  in  reduction  three  decimal  places  must  be 


052535  cu.  M. 


Equivalents 

1 meter  ......  ...=  39.37  inches.  1 kilometer = | miles. 

1 cubic  decimeter  = 1 liter.  1 liter  =1  quart. 

1 liter.  . = 1 kilogram.  1 kilogram = 2£  lbs. 

1 five  cent  piece  weighs  5 grams.  1 metric  ton.  ...  = 2204-6  lbs. 


1 hektare  = nearly  2\  Acres  or  2.47  Acres. 


ADDITION,  SUBTRACTION,  MULTIPLICATION,  DIVISION, 
in  the  Metric  System  is  the  same  as  for  whole  numbers  and  decimals  in  our  system. 

Reduce  the  different  denominations  all  to  the  same  denomination  by  moving  the 
decimal  point  or  annexing  ciphers,  and  perform  the  operation  indicated  by  the  signs. 

Reduction— Changing  from  one  table  to  another,  or  from  the  metric  system  to  the 
common  system,  use  the  above  equivalents. 

Example — 1,  What  will  3 cubic  meters  of  water  weigh? 

1 cu.  dm = 1 iiter.  1 liter = 1 kilogram. 

hence  3 cu.  m . 3000  cu.  dm.,  and 

3000  cu.  dm.  . = 3000 liters,  and  3000  liters  . . . . = 3000  kilogram.  Answer, 
or  if  we  wish  to  reduce  it  to  our  system  we  know,  1 kilogram  : . . = lbs. 

600  11 

3000  kg.  will  weigh  2\  times  as  much,  or  X = 6600  lbs.  Answer. 

2.  How  many  lbs.  will  a person  weigh  if  he  weighs  45  kilos? 

* 1 kilogram . . . = 2\  lbs. 

9 11 

45  kilos. = 45  X = 99  lbs.  Answer. 

3.  How  many  5 cent  pieces  in  2 kilograms? 

one  5 cent  piece  = 5 grams. 

2 Kg = 2000  grams. 

2000  -i-  5 s 400  five  cent  pieces. 

By  specific  gravity  of  any  material  is  meant  the  number  of  times  heavier  a sub- 
stance is  in  air  than  an  equal  volume  of  water. 

4.  If  4 liters  of  milk  weighs  4-12  Kg.  what  is  its  Specific  Gravity? 

(water),  1 1.  = 1 kg. 

(milk).  .1  1.  = i of  4.12  or  1.03  Kg. 

Weight  of  milk  is  1.03  times  that  of  water,  so  the  Specific  Gravity  is  1.03.  Ans. 


PAPERING 

A single  roll  of  wall  paper  is  24  feet  long  by  1|  feet  wide. 

A double  roll  is  48  feet  long  and  \\  feet  wide. 

A part  of  a roll  cannot  be  bought,  and  prices  are  always  quoted  by  the  single  roll. 
Borders  are  sold  by  the  yard  in  length. 

Rule : 

1.  Find  in  Jeet  the  perimeter  of  the  room  by  adding  the  length  of  the  4 walls; 
divide  this  by  1^  feet,  the  width  of  a roll,  to  find  how  many  strips  are  needed.  If 
there  is  any  remainder  count  it  an  extra  strip. 

2.  Divide  the  length  of  1 roll,  24  feet  if  single,  or  48  feet  if  double,  by  the  height 


ARITHMETIC— PART  I 


31 


of  the  room  in  feet , and  this  will  give  the  number  of  strips  that  can  be  cut  from  1 roll. 
If  there  is  any  remainder,  reject  it. 

3.  Divide  the  number  of  strips  required  for  the  room  by  the  number  obtained 
from  1 roll  and  the  result  will  be  the  number  of  rolls  required. 

Example — How  many  single  rolls  of  paper  will  it  take  for  a room  45  feet  by  33^ 
feet  and  11  \ feet  high? 

45  X 2 = 90  ft.  — 2 long  walls 
(a)  33f  X 2 = 67|  ft.  — 2 short  walls 
157|  ft.  — Perimeter. 


105 

m ? 24  2 ‘ 48  2 

(b)  157|  1|  = -j  X -j-  = 105  strips,  (c)  24  -f-  11J  = y X = 23  = 2 23  or 

2 strips  from  each  roll. 

105  strips  required  -4-  2 strips  from  1 roll  = 52|  rolls,  or  53  rolls  must  be  bought.  Ans. 

Note — Dealers  use  36  square  feet  as  the  area  of  a single  roll,  and  then  find  the 
area  of  all  the  walls  in  square  feet,  subtracting  from  this  the  area  of  all  openings, 
dividing  by  36,  the  area  of  the  single  roll,  to  obtain  the  number  of  rolls  required. 
The  area  of  the  ceiling  is  also  divided  by  36  to  obtain  the  number  of  single  rolls  re- 
quired for  it.  If  an  allowance  is  made  for  waste  in  matching  paper,  then  30  square 
feet  instead  of  36  are  used  as  the  single  roll 


PAINTING,  PLASTERING,  KALSOMINING 


Find  the  area  of  all  the  surfaces  to  be  covered,  by  multiplying  the  length  by  the 
width  of  each.  If  any  allowance  is  made  for  doors,  windows,  etc.,  find  their  area  in 
the  same  way  and  deduct  this  from  the  whole  area. 

Laths  are  4 feet  long  and  come  usually  in  bundles  of  1000,  no  less  than  a bundle 
can  be  bought. 

Example — What  will  it  cost  to  paint  the  walls  and  ceiling  of  a room  24  feet  by  15 
feet  by  10  feet  high  at  10  cents  a square  foot,  allowing  for  2 windows  6 ft.  X 3 ft.. 


and  for  one  door  7 ft.  X 3 ft.  ? 

Ceiling— 24  ft.  X 15  ft 360  sq.  ft. 

2 Long  Walls— 24  ft.  X 10  ft.  X 2.  . . . . = 480  sq.  ft. 

2 Short  Walls— 15  ft.  X 10  ft.  X 2.  = 300  sq.  ft. 


1.140  sq.  ft. 

Area,  2 Windows  (6  ft.  X 3 ft.)  X 2 = 36  sq.  ft.  ) _ r7  .. 

Area  Door  (7ft.  X 3 ft. ) = 21  sq.  ft.  sq‘ 

1,083  sq.  ft. 


$.10X1083  = $10.83 
$10.83  Answer. 


CARPETING 

Rule — Find  the  number  of  strips  required  by  dividing  the  width  of  the  room  if 
laid  lengthwise,  or  the  length  of  the  room  if  laid  crosswise,  by  the  width  of  1 strip  of 
carpet.  If  the  result  has  a fraction  count  it  an  extra  strip — for  if  the  exact  number  of 
strips  is  too  wide,  a part  of  one  breadth  or  strip  is  turned  under. 

In  matching  figures  the  ends  of  the  strips  after  the  first  strip,  will  sometimes  have 
to  be- turned  under. or  cut  off,  thereby  necessitating  more  carpet  than  is  called  for  by 
the  actual  size  of  the  room. 

Multiply  the  length  of  a strip  required  for  the  room  by  the  number  of  strips  and 
the  result  will  be  the  quantity  required. 

Example — How  many  yards  carpet  f yard  wide  laid  lengthwise  will  be  required 
to  cover  a floor  16  feet  by  14  feet? 

Width  of  room  14  ft.  -f-  3 = 4§  yards. 

Width  of  room  4§  yds.  -s-  width  of  carpet  | yds.  = *§*  Xf  = 5§6  = 6|  or  7 strips. 

One  strip  the  length  of  the  room  is  16  feet  or  5£  yards ; 5£  yds.  X 7 the  number  of 
strips  = Jg6  X 7 = 1i-  .=  37^  yds.  Answer. 


ROOFING  AND  FLOORING 

This  is  usually  done  by  the  square , meaning  100  square  feet. 

1000  shingles  are  usually  allowed  to  a square.  Shingles  are  sold  by  the  bundle — 
meaning— 1000  shingles  measuring  18"  by  4",  or  by  the  bunch  meaning  250  shingles. 
No  less  than  a bunch  can  be  bought. 

Sometimes  roofing  and  flooring  is  done  by  the  square  foot  or  square  yard. 


32 


3 0112  105669409 

ARITHMETIC -PART  I 


Example — How  much  would  it  cost  to  shingle  both  sides  of  a 40  foot  roof  measur- 
ing from  eaves  to  ridge  25  feet  at  $6.75  a thousand? 

Solution— 2 sides  roof  = 40  ft.  X 25  ft.  X 2 = 2,000  square  feet  (for  every  100 
square  feet,  1 bundle  or  1000  shingles  will  be  required. ) . \ 2000  sq.  ft.  -r-  100  sq.  ft. 

= 20  bundles.  Each  bundle  of  1000  cost  $6-75,  therefore  20  will  cost 

5 

3 27  W 


$6 


X 20=  j- 


X 


= $135.  Answer. 


LUMBER 

A board  foot  is  the  unit  of  measure  and  is  a board  one  foot  long , one  foot  wide 
and  one  inch  thick. 

When  lumber  is  1 inch  or  less  thick,  to  find  the  number  board  feet,  multiply  the 
length  of  the  board  by  the  width.  If  it  is  more  than  an  inch  then  multiply  the  length 
in  feet  by  the  width  in  feet  and  that  result  by  the  number  of  inches  thick.  Therefore 
a board  10  feet  by  6 inches  by  2 inches  would  contain  10'  X I1  X 2"  = 10  board  feet. 

CAPACITY  OF  CISTERNS 

Rule — Divide  the  contents  of  the  cistern  in  cubic  inches  by  231,  to  find  the  number 
of  gallons. 

Example — 1.  How  many,  cubic  feet  in  a cistern  containing  50  hogsheads? 

Solution — 50  hhds  X 63  = 3150  gallons.  As  231  cubic  inches  = 1 gallon,  in  3150 
gallons  we  will  have  231  X 3,150  or  727,650  cubic  inches 

There  are  1728  cubic  inches  in  each  cubic  foot,  so  727650  -4-  1728  = 421gVi  cubic 
feet.  Answer. 

2.  How  many  gallons  in  a tank  12  feet  by  5 feet  by  8 feet? 
a.  12  ft.  X 5 ft.  X 8 ft.  = 480  cubic  feet.  b.  480  cu.  it.  X 1728  = 829440  cu.  in. 

829440  cubic  inches  231  = 3,590f-f  gallons.  Answer. 

CAPACITY  OF  BINS 

Rule — Divide  the  contents  of  the  bin  in  cubic  inches  by  2150.42  to  find  the  number 
of  bushels,  for  1 bushel  ==  2150.42  cubic  inches. 

Example  1.  How  many  bushels  in  a bin  12  feet  by  5 feet  by  8 feet? 

12  ft.  X 5 ft.  X 8 ft.  = 480  cubic  feet.  480  X 1728  = 829440  cubic  inches. 

76415 

829440  -4-  2150. 42  = 385  mb2l~  bushels. 

The  above  gives  the  exact  result,  but  ordinarily  the  following  method  gives 
accurate  enough  results  and  is  easier. 

Rule — Divide  contents  of  the  bin  in  cubic  feet  by  1*  cubic  feet  to  find  the  number 
of  bushels. 

2150.42  cubic  inches  (in  1 bushel)  is  to  1728  cubic  inches  (In.  1 cu.  ft.)  (very  nearly) 
as  5 is  to  4 ; so  1 bushel  = ® or  1*  cubic  feet. 

Example  above  according  to  this  method  would  be  solved. 

96  4 

12  X 5 X 8 = 480  cubic  feet.  480  -4-  1}  = X = 384  bushels.  Ans. 

THERMOMETERS 

In  Centigrade  thermometers  the  freezing  point  is  0°  and  the  boiling  points  100°, 
giving  100°  — 0°  or  100°. 

In  Fahrenheit  thermometers  the  freezing  points  32°  and  the  boiling  point  212°. 

2123  — 32°  = 180°  number  degrees  between  freezing  and  boiling  on  the  Fahrenheit 
thermometer. 

100  5Q 

Therefore  180°  Fahrenheit  = 100°  Centigrade,  hence  1°  Fahrenheit  = ^ or  -g- 
Centigrade.” 

Example — What  degree  on  the  Centigrade  thermometer  corresponds  to  68°  on  the 
Fahrenheit? 

4 5 

68°  — 32°  = 36°  Fahrenheit.  $$  X — = 20°  Centigrade.  Answer. 

1 


